On intrinsic ergodicity of shift spaces with(almost) weak specification

Dominik Kwietniakjoint work with

Piotr Oprocha (AGH) Michał Rams (IM PAN)

Santiago de Chile December 18 2014

Example(s)

There exists a family of shift spaces which contains

I A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1

I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1

I A shift space with almost specification but without weakspecification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which contains

I A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1

I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1

I A shift space with almost specification but without weakspecification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1

I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1

I A shift space with almost specification but without weakspecification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1

I A shift space with almost specification but without weakspecification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specification

I An intrinsically ergodic shift space X and its factor Y withmultiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Example(s)

There exists a family of shift spaces which containsI A shift space with almost specification and multiple MMEs1I A shift space with weak specification and multiple MMEs1I A shift space with almost specification but without weak

specificationI An intrinsically ergodic shift space X and its factor Y with

multiple MMEs such that there are Climenhaga-Thompsondecompositions for X and Y respectively

B(X ) = CpX middot GX middot CsX

B(Y ) = CpY middot GY middot CsY

with h(GX ) gt h(CpX cupCsX) and h(GY ) lt h(CpY cupCsY)

1Proved independently by Ronnie Pavlov

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Definition(s)

A shift space X has the

I almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Definition(s)

A shift space X has theI almost specification property if there exists θ Nrarr N withθ isin o(n) such that for every n isin N and w1 wn isin B(X )there exist words v1 vn isin B(X ) with |vi | = |wi | such thatv1v2 vn isin B(X ) and each vi differs from wi in at mostθ(|vi |) places

I weak specification property if there exists t Nrarr N witht isin o(n) such that for any words u isin B(X ) w isin Bn(X ) thereexists a word v isin Bt(n)(X ) such that x = uvw isin B(X )

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Climenhaga-Thompson decompositions

We say that the language B(X ) of a shift space X admitsClimenhaga-Thompson decomposition if there areCpG Cs sub B(X ) such that

(I) for every w isin B(X ) there are up isin Cp v isin G us isin Cs suchthat w = upvus

(II) there exists t isin N such that for any n isin N andw1 wn isin G there exist v1 vnminus1 isin B(X ) such thatx = w1v1w2v2 vnminus1wn isin B(X ) and |vi | = t fori = 1 n minus 1

(III) For every M isin N there exists τ isin N such that givenw isin B(X ) satisfying w = upvus for some up isin Cp v isin Gus isin Cs with |up| not M and |us | not M there exist wordsuprime uprimeprime with |uprime| not τ |uprimeprime| not τ for which uprimewuprimeprime isin G

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Results of Climenhaga amp Thompson

Theorem (Climenhaga amp Thompson 2012)If a shift space X has the CT decomposition given by the sets CpG Cs and

h(G) gt h(Cp cupCs)

then X is intrinsically ergodic Furthermore if

h(Cp cupCs) = 0

then every shift factor of X is intrinsically ergodic

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X

X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

[PS]X

X X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X X

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like notions

per specification specification almost specification

per weak spec weak specification approx prod structure

X

X

X[PS]

X X

X

X[KOR]

X

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Positive results

Theorem (lArrKulczycki DK Oprocha rArrChen Oprocha Wu)A dynamical system has the almost specification property if andonly if its restriction to the measure center does

Theorem (DK Oprocha Rams)The restriction of a dynamical system with the almost specificationproperty to its measure center is a topological K system withdense set of minimal points In particular this restriction istopologically weakly mixing and non-minimal provided that themeasure center is non-trivial

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2

I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)

I Identify (a b) isin A withlfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the color

I The symbol 0 =lfloor00

rceilis the marker

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR

I Fix p q isin N p q shy 2I Pick R = Rninfinn=1 such that

1 = R1 sub R2 sub R3 sub and Rn sub 1 n

Assume also that for each k there is n such that1 n Rn contains k consecutive integers

I Let r N0 rarr N0 by r(0) = 0 and

r(n) = |Rn| for n isin N

I Let A = 1 p times 0 1 q minus 1 cup (0 0)I Identify (a b) isin A with

lfloorab

rceiland regard a isin 0 1 p as

the colorI The symbol 0 =

lfloor00

rceilis the marker

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk

I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin R

I A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocks

I A word W is allowed if there exist U isinM andV1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Construction of XR cont

I A wordlfloora1anb1bn

rceilisin Alowast is monochromatic or of color

a isin 0 1 p if a = a1 = = an and polychromaticotherwise

I A monochromatic block W =lfloora1anb1bn

rceilisin Alowast is restricted if

bj = 0 for each j isin Rk I A block W is free if W = 0V where V isin RI A good word is finite concatenation of free blocksI A word W is allowed if there exist U isinM and

V1 Vk isin F such that W = UV1 Vk The set of allallowed words is the language of XR

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Almost (weak) specification of XR

Lemma

1 If r(n)nrarr 0 as nrarrinfin then the shift space XR has thealmost specification property

2 Let Nk denote the smallest n such that 1 n Rn

contains k consecutive integers The shift space XR has theweak specification property if and only if kNk rarr 1 ask rarrinfin

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin

In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

A technical lemma

LemmaIf r(n) gt 0 for every n and lim inf

nrarrinfinr(n)

ln ngt 0 then there is Q shy 2

such that the seriesinfinsumn=1

qminusr(n) converges for all integers q shy Q and

its sum tends to 0 as q rarrinfin In particular for every p shy 2 there isq shy Q such that

1 + pinfinsumj=1

qminusr(j) not q (1)

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Counting words

LemmaLet Fn (Gn) be the number of n-blocks in F (G) Then

1 F0 = F1 = 1 and Fn = p middot q(nminus1)minusr(nminus1) for all n gt 1

2 G0 = G1 = 1 and

Gn =nsum

i=1

F i Gnminusi = Gnminus1+nminus1sumj=1

pqjminusr(j) Gnminus1minusj for n gt 1

Lemma

1 If (1) holds then Gn not qn for every n shy 0 In particularh(XR) = log q and XR has at least p MMErsquos

2 If (1) does not hold then

lim infnrarrinfin

log Gnn

gt log q

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

CT decomposition and intrinsic ergodicity of XR

LemmaLet Cp =M Cs = empty Then CsG Cp is a Climenhaga-Thompsondecomposition for XR and (1) does not hold if and only if

h(G) gt h(Cs cupCp) = h(Cp)

Therefore XR s intrinsically ergodic if and only if (1) does not hold

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Orbit segments and specifications

I Let (X ρ) be a compact metric space

I Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuous

I Let a b isin N cup 0 a not b The orbit segment of x isin X over[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Orbit segments and specifications

I Let (X ρ) be a compact metric spaceI Let T X rarr X be continuousI Let a b isin N cup 0 a not b The orbit segment of x isin X over

[a b] is the sequence

T [ab](x) = (T a(x)T a+1(x) T b(x))

I A specification is a family of orbit segments

ξ = T [aj bj ](xj)nj=1

such that n isin N cup infin and bj lt aj+1 for all 1 not j lt n

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like properties

I A ldquogenericrdquo specification property

There exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications

such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification-like properties

I A ldquogenericrdquo specification propertyThere exists a family Ξ of ldquogoodrdquo specifications such thatevery specification from Ξ can approximated by an orbit of asingle point in X

I Varying the notion of ldquogood specificationrdquo andldquoapproximationrdquo we obtain various specification-likeproperties

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification property

I Let N isin N

A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Specification property

I Let N isin N A family of orbit segments ξ = T [aj bj ](xj)nj=1 isan N-spaced specification if ai minus biminus1 shy N for 2 not i not n

I We say that a specification ξ = T [aj bj ](xj)nj=1 is ε-traced byy isin X if

ρ(T k(y)T k(xi )) not ε for ai not k not bi and 1 not i not n

I We say that T has the specification property if for any ε gt 0there is N = N(ε) isin N such that any N(ε)-spacedspecification is ε-traced by some y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Weak specification

I Let L Nrarr N be a function

A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Weak specification

I Let L Nrarr N be a function A family of orbit segmentsT [aj bj ](xj)nj=1 is an L-spaced specification if

ai minus biminus1 shy L(bi minus ai ) for 2 not i not n

I We say that T has the weak specification property if for anyε gt 0 there is a function Lε Nrarr N with Lε(m)mrarr 0 as(mrarrinfin) such that any Lε-spaced specification is ε-traced bysome y isin X

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Bowen balls with mistakes

I A nondecreasing map g Nrarr N is called a blowup function iffor all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Bowen balls with mistakesI A nondecreasing map g Nrarr N is called a blowup function if

for all n isin N one has g(n) lt n and

limnrarrinfin

g(n)n = 0

I Given a blowup function g we define the functionkg R+ rarr N such that kg (ε) is the smallest integer n suchthat g(m) lt εm for all m shy n and we define the set

P(n g) =

Λ sub 〈n〉 |〈n〉 Λ| not g(n)

I For Λ sub 0 1 2 and x y isin X we define

ρΛ(x y) = maxjisinΛρ(T j(x)T j(y)

I For every x isin X and ε gt 0 let

Bn(g x ε) =y isin X minρΛ(x y) Λ isin P(n g) not ε

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Almost specification property

Definition (Pfister amp Sullivan)A dynamical system (X T ) satisfies the almost specificationproperty if there exists a blowup function g such that for anyk isin N ε1 εk gt 0 any n1 shy kg (ε1) nk shy kg (εk) andevery specification ξ = T [ajminus1aj )(xj)kj=1 such that a0 = 0 andaj minus ajminus1 = nj one can find a point y in

k⋂j=1

Tminusajminus1Bnj (g xj εj) 6= empty

We say that y (g nj εj)-traces xj over [ajminus1 aj)

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Approximate product structure

Definition (Pfister amp Sullivan)We say that T has the approximate product structure if for anyε gt 0 δ1 gt 0 and δ2 gt 0 there exists an integer N gt 0 such thatfor any n shy N and any sequence xiinfini=1 of X there exist asequence of integers hiinfini=1 and a point y isin X satisfying h1 = 0n not hi+1 minus hi not n(1 + δ2) and∣∣∣0 not j lt n ρ

(T hi+j(y)T j(xi )

)gt ε

∣∣∣ not δ1n for any i isin N

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817

Some references I

Marcin Kulczycki Dominik Kwietniak Piotr OprochaOn almost specification and average shadowing propertiesFund Math 224 (2014) 241ndash278 doi104064fm224-3-4

Dominik Kwietniak Piotr Oprocha Michał RamsOn entropy of dynamical systems with almost specificationarXiv14111989[mathDS]

Ronnie PavlovOn intrinsic ergodicity and weakenings of the specificationpropertyarXiv14112077[mathDS]

V Climenhaga D ThompsonIntrinsic ergodicity beyond specification β-shifts S-gap shiftsand their factorsIsrael J Math 192 (2012) 785ndash817