MULTIFRACTAL ANALYSIS FOR QUOTIENTS OF BIRKHOFF …giommi/IMRN_iommi_jordan_revised.pdf ·...

$: MULTIFRACTAL ANALYSIS FOR QUOTIENTS OF BIRKHOFF …giommi/IMRN_iommi_jordan_revised.pdf · GODOFREDO IOMMI AND THOMAS JORDAN Abstract. This paper is devoted to study multifractal$
MULTIFRACTAL ANALYSIS FOR QUOTIENTS OF BIRKHOFFSUMS FOR COUNTABLE MARKOV MAPS

GODOFREDO IOMMI AND THOMAS JORDAN

Abstract. This paper is devoted to study multifractal analysis of quotients ofBirkho! averages for countable Markov maps. We prove a variational principlefor the Hausdor! dimension of the level sets. Under certain assumptions weare able to show that the spectrum varies analytically in parts of its domain.We apply our results to show that the Birkho! spectrum for the Manneville-Pomeau map can be discontinuous, showing the remarkable di!erences withthe uniformly hyperbolic setting. We also obtain results describing the Birkho!spectrum of suspension flows. Examples involving continued fractions are alsogiven.

1. Introduction

Multifractal analysis is a branch of the dimension theory of dynamical systems.It typically involves decomposing the phase space into level sets where some localquantity takes a fixed value, say a. The standard problems are to find the Hausdor!dimension of these sets and to determine how the dimension varies with the param-eter a. In hyperbolic dynamical systems there are several local quantities whichcan be studied in such a way. These quantities are dynamically defined, examplesare local dimensions of Gibbs measures, Birkho! averages of continuous functionsand local entropies of Gibbs measures. In this setting we normally find two typesof results, on the one hand variational principles are obtained to determine thedimension of the level sets and on the other thermodynamic formalism is used toprove that the spectra varies in an analytic way. See [Ba] for an overview of someof these results.

In this note we consider the multifractal analysis for quotients of Birkho! aver-ages for a countable branch Markov map. In the case of expanding finite branchMarkov maps on the interval the multifractal analysis for Birkho! averages or quo-tients of Birkho! averages of continuous functions is well understood, for examplesee [BS1, FLW, O, FLP, C]. In particular, if the Markov map is expanding, C1+!

and the continuous functions are Holder the multifractal spectra vary analytically[BS1]. However, it turns out that substantial di!erences can occur in the case wherethere are countably many branches and Birkho! averages are studied, note that thespace is no longer compact. In this setting, phase transitions may occur, for ex-ample see the work in [FJLR, IJ, KMS]. Nevertheless, the dimension of sets in the

Date: August 14, 2013.G.I. was partially supported by the Center of Dynamical Systems and Related Fields codigo

ACT1103 and by Proyecto Fondecyt 1110040. The work on this project was started during T.J.’svisit to Santiago supported by Proyecto Mecesup-0711. We would also like to thank LingminLiao and Micha"l Rams for helpful comments. Finally, we wish to thank the referee for all theinteresting remarks and comments.

1

2 GODOFREDO IOMMI AND THOMAS JORDAN

multifractal decomposition can still generally be found by a conditional variationalprinciple. Our aim is to extend these results to fairly general quotients of Birkho!sums and with additional assumptions examine the smoothness of the multifractalspectra. Note that the multifractal analysis of quotients of Birkho! sums has al-ready been studied in [KU] but with the assumption that the denominator is theLyapunov exponent.

One motivation for our work is that studying quotients of Birkho! averages for acountable branch Markov map can relate to studying Birkho! averages for a finitebranch non-uniformly expanding map. In [JJOP] multifractal analysis for Birkho!on finite branch non-uniformly expanding maps was studied, while a conditionalvariational principle was obtained the question of how the spectra varied was notfully addressed. By tackling this problem via countable state expanding maps weare able to obtain new results for Holder functions in this setting and to showthat while in some regions the spectrum varies analytically it is also possible forit to have discontinuities. Another motivation is to study related problems forsuspension flows where Birkho! averages on the flow correspond to quotients ofBirkho! averages for the base map.

Let us be more precise and define the dynamical systems under consideration.Denote by I = [0, 1] the unit interval. As in [IJ] we consider the class of EMR(expanding-Markov-Renyi) interval maps.

Definition 1.1. Let {Ii}i!N be a countable collection of closed intervals wherewhere int(Ii) ! int(Ij) = ! for i, j " N with i #= j and Ii $ I for every i " N. Amap T : %"n=1In & I is an EMR map, if the following properties are satisfied

1. The intervals in the partition are ordered in the sense that for every i " Nwe have sup{x : x " Ii+1} ' inf{x : x " Ii+1}. Moreover, zero is the uniqueaccumulation point of the set of endpoints of {Ii}.

2. The map is C2 on %"i=1int Ii.3. There exists ! > 1 and N " N such that for every x " %"i=1Ii and n ( N we

have |(Tn)#(x)| > !n.4. The map T is Markov and it can be coded by a full-shift on a countable

alphabet.5. The map satisfies the Renyi condition, that is, there exists a positive number

K > 0 such that

supn!N

supx,y,z!In

|T ##(x)||T #(y)||T #(z)| ' K.

The repeller of such a map is defined by

" := {x " %"i=1Ii : Tn(x) is well defined for every n " N} .

The Markov structure assumed for EMR maps T , allows for a good symbolic rep-resentation (see Section 2.1).

Example 1.2. The Gauss map G : (0, 1] & (0, 1] defined by

G(x) =1x)

! 1x

",

where [·] is the integer part, is a standard example of an EMR map.

MULTIFRACTAL ANALYSIS FOR QUOTIENTS OF BIRKHOFF SUMS 3

For " " R and # " R" (see Definition 2.4 for a precise definition of the class ofpotentials R and R" ) we will denote

$m = inf

#lim

n$"

$n%1i=0 "(T ix)

$n%1i=0 #(T ix)

: x " "

%and

$M = sup

#lim

n$"

$n%1i=0 "(T ix)

$n%1i=0 #(T ix)

: x " "

%.

Throughout the paper we will assume that $m #= $M since otherwise our resultsbecome trivial. Note that, since the space " is not compact, it is possible for $M

to be infinity. For $ " [$m, $M ] we define the level set of points having Birkho!ratio equal to $ by

J($) =

#x " " : lim

n$"

$n%1i=0 "(T ix)

$n%1i=0 #(T ix)

= $

%.

Note that these sets induce the so called multifractal decomposition of the repeller,

" =#M&

#=#m

J($)&

J #,

where J # is the irregular set defined by,

J # =

#x " " : the limit lim

n$"

$n%1i=0 "(T ix)

$n%1i=0 #(T ix)

does not exist

%.

The multifractal spectrum is the function that encodes this decomposition and it isdefined by

b($) = dimH(J($)),where dimH(·) denotes the Hausdor! dimension (see Subsection 2.2). The form ofour results depend upon the value of $. We will split the open interval ($m, $M )into two subsets. Firstly we define

$ = inf'

$ " R : there exists {xn}n!N where limn$"

xn = 0 and limn$"

"(xn)#(xn)

= $

(

and

$ = sup'

$ " R : there exists {xn}n!N where limn$"

xn = 0 and limn$"

"(xn)#(xn)

= $

(.

We let E = [$,$], U = ($m, $M )\E and M(T ) be the space of all T -invariantprobability measures for which # and log |T #| are integrable. We can now stateour first result. In our first theorem we establish a variational principle for thedimension of level sets.

Theorem 1.3. If " " R and # " R" then for all $ " U

b($) := dimHJ($) = supµ!M(T )

'h(µ)%(µ)

:)

"dµ)#dµ

= $

(

and for $ " E ! ($m, $M )

b($) := dimHJ($) = lim!$0

supµ!M(T )

'h(µ)%(µ)

:)

"dµ)#dµ

" ($) &, $ + &)(

.


Here %(µ) is the Lyapunov exponent and M(T ) is the space of T)invariantmeasures (see section 2.1 for precise definitions). A particular case we will beinterested is when the function # satisfies that limx$0

$(x)log |T !(x)| = * and also "

and # are such that $M < *. With these additional assumptions we are able tosay more about the smoothness of the function $ & dimHJ($).

Theorem 1.4. Let " " R and # " R" be such that

limx$0

#(x)log |T #(x)| = *,

and that $M < *. There then exists three pairwise disjoint intervals J1, J2, J3 suchthat

1. J1 % J2 % J3 = ($m, $M ),2. J1 ' J2 ' J3

3. The function $ & dimHX# is analytic on J1 and J3

4. For $ " J2, dimHX# = dimH"5. It is possible that J1 = !, J3 = ! or that J2 is a single point.

The rest of the paper is laid out as follows. In section 2 we give some preliminaryresults necessary for the rest of paper. In sections 3 and 4 respectively the upperand lower bounds for Theorem 1.3 are proved. In section 5 Theorem 1.4 is proved,section 6 shows there may be examples with discontinuities in the spectrum andgives applications to non-uniformly expanding maps. Section 7 looks at the case ofsuspension flows and finally in section 8 we provide examples coming form numbertheory.

2. Preliminaries

This section is devoted to provide the necessary tools and definitions that willbe used in the rest of the paper.

2.1. Thermodynamic Formalism for EMR maps. In order to define the ther-modynamic quantities and to establish their properties for an EMR map we willmake use of the analogous theory developed at a symbolic level. Let N be thecountable alphabet, the full-shift is the pair (#, ') where # = {(xi)i&1 : xi " N} ,and ' : # & # is the shift map defined by '(x1x2 · · · ) = (x2x3 · · · ). We equip #with the topology generated by the cylinders sets

Ci1···in = {x " # : xj = ij for 1 ' j ' n}.

The Markov structure assumed in the definition of EMR map implies that thereexists a continuous map, the natural projection, ( : # & " such that ( +' = T +(.Moreover the map ( : # & " \

*n!N T%nE is surjective and injective except on at

most a countable set of points. Denote by I(i1, . . . in) = ((Ci1...in) the cylinder oflength n for T . For a function f " # and n ( 1 we will define the n)variations off by

varn(f) = sup(i1,...,in)!Nn

supx,y!Ci1···in

|f(x)) f(y)|

and say that f is locally Holder if there exists 0 < ) < 1 and A > 0 such that for alln ( 1 we have varn(f) ' A)n. We now define the main object in thermodynamicformalism,


Definition 2.1. The topological pressure of a potential " : " & R such that " + (is locally Holder is defined by

PT (") = sup'

h(µ) ++

" dµ : )+

" dµ < * and µ "MT

(,

where MT denotes the space of T)invariant probability measures. A measure at-taining the supremum is called an equilibrium measure for ".

The following definition of pressure (at a symbolic level) is due to Mauldin andUrbanski [MU1],

Definition 2.2. Let " : # & R be a potential of summable variations, the pressureof " is defined by

(1) P%(") = limn$"

1n

log,

%n(x)=x

exp

-n%1,

i=0

"('ix)

..

The above limit always exits, but it can be infinity. The next proposition re-lates these two notions and allows us to translate results obtained by Mauldin andUrbanski [MU1, MU2] and by Sarig [Sa1, Sa2, Sa3] to our setting. For n " N wewill denote

#n = {x " # : xi ' n}and "n = ((#n). Note that "n is a T -invariant set.

Proposition 2.3. Let T be an EMR map. If " : " & R such that $ = " + ( islocally Holder then

1.PT (") = P%($).

2. (Approximation property.)

P (") = sup{P%|K($) : K $ (0, 1] : K #= ! compact and '-invariant},

where P%|K(") is the classical topological pressure on K (for a precise defi-nition see [W, Chapter 9]). In particular

P (") = supn!N

{PT |!n(")}.

3. (Regularity.) If P (") < * then there exists a critical value t' " (0, 1] suchthat for every t < t'we have that P (t") = * and for every t > t'we havethat P (t") < *. Moreover, if t > t' then the function t & P (t") is realanalytic, strictly convex and every potential t" has an unique equilibriummeasure. Moreover the function t & PT |!n

(t") is analytic and convex forall t " R.

Note that if T is an EMR map then the potential log |T #| + ( is locally Holderand P () log |T #|) < *. If µ "MT then the integral

%(µ) :=+

log |T #| dµ,

will be called the Lyapunov exponent of µ. Of particular interest will be the fol-lowing classes of potentials,


Definition 2.4. We define the following collections of potentials

R = {" : " & R : " is uniformly bounded below and " + ( is locally Holder}

and for * > 0

R" = {" " R : for every x " " we have "(x) ( *} .

2.2. Hausdor! Dimension. In this subsection we recall basic definitions fromdimension theory. We refer to the books [Ba, Fa] for further details. A countablecollection of sets {Ui}i!N is called a +-cover of F $ R if F $

*i!N Ui, and for every

i " N the sets Ui have diameter |Ui| at most +. Let s > 0, we define

Hs& (F ) := inf

# ",

i=1

|Ui|s : {Ui}i is a +-cover of F

%

andHs(F ) := lim

&$0Hs

& (F ).

The Hausdor! dimension of the set F is defined by

dimH(F ) := inf {s > 0 : Hs(F ) = 0} .

We will also define the Hausdor! dimension of a probability measure µ by

dimH(µ) := inf {dimH(Z) : µ(Z) = 1} .

3. Proof of Upper bound of Theorem 1.3

Throughout this section we will let let " " R and # " R". We wish to provethat

(2) dimHJ($) ' supµ!M(T )

'h(µ)%(µ)

:)

"dµ)#dµ

= $

(:= +($).

However to prove this we will need that $ " U . When $ /" U , in general, we canonly show that

(3) dimHJ($) ' lim!$0

supµ!M(T )

'h(µ)%(µ)

:////

)"dµ)#dµ

) $

//// ' &

(.

Our method will be to prove (3) for all $ " ($m, $M ) and then to deduce (2) for$ " U by showing that + is continuous in this region. We first show the continuityof +($) when $ " U . To do this we need the following preparatory Lemma aboutthe set U . This Lemma will also be used in the proof of Theorem 1.4. Denote byB($, )) the ball of center $ and radius ).

Lemma 3.1. For any $ " U there exists ) > 0 and C1 > 0 such that if µ " MT

withR

'dµR$dµ

" B($, )) then)

#dµ ' C1.

Proof. We will let $ " U and assume that $ < $, since the case when $ > $ canbe treated analogously. We let ) = #%#

3 and note that by the definition of U thereexists y# " (0, 1) such that if x ' y# then "(x) ( ($ + 2))#(x). We also let C2 " Rbe such that

max'

supx&y!

{|"(x)|}, supx&y!

{|#(x)|}(' C2.


Assume that µ is a T -invariant probability measure such that+

#dµ < * and)

"dµ)#dµ

" B($, )).

We can estimate

($ + ))+

#dµ (+

"dµ (+

x&y!"(x)dµ + ($ + 2))

+

x(y!#(x)dµ

and rearrange to get

))

+

x(y!#(x)dµ + ($ + ))

+

x&y!#(x)dµ)

+

x&y!"(x)dµ ( 0.

Thus, using the definition of C2 we obtain

)

+

x(y!#(x)dµ ' C2|$ + ) + 1|.

Therefore, +#dµ ' C2

0|$ + ) + 1|

)+ 1

1

which completes the proof. !

Lemma 3.2. The function + : ($m, $M ) & R defined by,

+($) := supµ!M(T )

'h(µ)%(µ)

:)

"dµ)#dµ

= $

(

is continuous on U .

Proof. Let $ " U . We will show that

lim inf!$0

inf{+()) : ) " ($) &, $ + &)} ( +($) and

lim sup!$0

sup{+()) : ) " ($) &, $ + &)} ' +($).

We can find two measures µ1, µ2 " MT such that %(µ1),)

#dµ1,)

#dµ2 < * and)

"dµ1)#dµ1

< $ <

)"dµ2)#dµ2

.

By Lemma 3.1 there also exists a constant K > 0 such that for any & > 0 we canfind a measure µ such that

)#dµ ' K,

R'dµ1R$dµ1

= $ and h(µ)((µ) ( +($)) &. To prove

that lim inf!$0 inf{+()) : ) " ($ ) &, $ + &)} ( +($) we simply need to considerthe following convex families of measures pµ + (1) p)µ1 and pµ + (1) p)µ2, wherep " (0, 1).

To show that lim sup!$0 sup{+()) : ) " ($ ) &, $ + &)} ' +($) we consider asequence of T -invariant measures (,n)n such that

limn$"

h(,n)%(,n)

( +($) and limn$"

)"d,n)#d,n

= $.

By considering the appropriate convex combination with either µ1 or µ2 we can nowfind a sequence (*n)n of T -invariant measures with

R'd"nR$d"n

= $ and limn$"h("n)(("n) (

+($). The result immediately follows. !


Let C > 0 be a constant such that

max

# ",

k=1

vark("),",

k=1

vark(#),",

k=1

vark(log |T #|)%' C.

We work in a similar way to [IJ]. Let Sk"(x) :=$k%1

i=0 "(T ix) and

J#,N,! :='

x " " :Sk"(x)Sk#(x)

" ($) &, $ + &) for every k ( N

(.

Note that for any & > 0 we have

J# $ %"N=1J($, N, &).

So we can obtain upper bounds of the dimension of the set J($) by obtaining upperbounds for the dimension of J($, N, &). For k ( N we will define covers by

Ck = {I(i1, . . . , ik) : I(i1, . . . , ik) ! J($, N, &) #= !}.In [IJ] analogous covers were defined in Section 3. However, in that setting thesecovers had finite cardinality whereas here the cardinality can be infinite which willcause additional di%culties. From now on $ and & > 0 will be fixed. We will let

tk := inf

23

4t " R :,

I(i1,...,ik)!Ck

|I(i1, . . . , ik)|t ' 1

56

7 .

A covering argument then gives that dimHJ($, N, &) ' lim supk$" tk. We wish torelate the values tk to T -invariant probability measures. We start with the followinglemma.

Lemma 3.3. There exists K " N such that for all k ( K, I(i1, . . . , ik) and x, y "I(i1, . . . , ik) we have

Sk"(x)Sk#(x)

) Sk"(y)Sk#(y)

' &.

Proof. By the assumption that both # and " are of summable variations we knowthat for all k " N.

Sk"(x)) C

Sk#(x) + C' Sk"(y)

Sk#(y)' Sk"(x) + C

Sk#(x)) C.

However, we have by assumption that Sk#(x) ( k* for all x " ". The result nowimmediately follows. !

We can now construct the measures we need.

Lemma 3.4. If t " R satisfies that,

I(i1,...,ik)!Ck

|I(i1, . . . , ik)|t > 1

then there exists a T -invariant probability measure µk such that)

"dµk)#dµk

" ($) 2&, $ + 2&)

andh(µk)%(µk)

( t)A(k)

where A(k) & 0 as k &*.


Proof. By the assumptions in our theorem it is possible to find a finite set Dk $ Ck

such that ,

I(i1,...,ik)!Dk

|I(i1, . . . , ik)|t = Zk > 1.

We can now construct a T k invariant Bernoulli measure, ,k by assigning eachcylinder in Dk weight 1

|Dk| |I(i1, . . . , ik)|t. We can now estimate

h(,k, T k)%(,k, T k)

=)

$I(i1,...,ik)!Dk

1|Dk| |I(i1, . . . , ik)|t log( 1

|Dk| |I(i1, . . . , ik)|t)%(,k, T k)

=)

$I(i1,...,ik)!Dk

1|Dk| |I(i1, . . . , ik)|t log |I(i1, . . . , ik)|t

%(,k, T k)+

Zk log |Dk||Dk|%(,k, T k)

( t%(,k, T k)) C

%(,k, T k)( t) C

k log !.

By Lemma 3.3 we have that for k su%ciently largeR

Sk'd)kRSk$d)k

" ($ ) 2&, $ + 2&).

To complete the proof we let µk = 1k

$k%1i=0 ,k + T%i and note that C

k log * & 0 ask &*. !

It now follows that for any + > 0 we can find a sequence of T -invariant measures(µk)k such that lim supk$"

///tk ) h(µk)((µk)

/// ' + and whereR

'dµkR$dµk

" ($ ) 2&, $ + 2&).Hence, for all & > 0 we have

dimHJ($) ' supµ!M(T )

'h(µ)%(µ)

:)

"dµ)#dµ

" ($) &, $ + &)(

.

To complete the proof of the upper bound for $ " U we simply apply Lemma 3.2.

4. Proof of the lower bound of Theorem 1.3

Our method to prove the lower bound is to find an ergodic measure µ withR'dµR$dµ

= $, or a sequence of ergodic measures µn with limn$"

R'dµnR$dµn

= $, and

then use the fact that the dimension of ergodic measures with finite entropy is h(µ)((µ) ,

see [MU2, Thereom 4.4.2]. For $ " U we will use the thermodynamic formalism toshow that there is an ergodic equilibrium measure with dimension +($), which willshow that dimHJ($) ( +($). For $ /" U we will need to work slightly harder.

We define the function G1 : R3 & R % {*} by

G1($, q, +) = P (q(") $#)) + log |T #|).Note that G1 can be infinite and we will adopt the convention * > 0. We have thefollowing lemma.

Lemma 4.1. Let $ " ($m, $M ) and + > 0. If for all q " R we have thatG1($, q, +) > 0 then dimHJ($) > +.

Proof. Since $ " ($m, $M ) it follows that

limq$"

G1($, q, +) = limq$%"

G1($, q, +) = *.

Indeed, this is a consequence of ergodic optimisation results that relate the asymp-totic derivative of the pressure to maximising/minimising measures for the poten-tial ") $# (see for instance [JMU, Theorem 1]). This means that if we let q% :=


inf{q " R : G1($, q, +) < *} and q+ := sup{q " R : G1($, q, +) < *} then eitherlimq$q" G($, q, +) = * or G($, q%, +) < * and similarly limq$q+ G($, q, +) = *or G($, q+, +) < *. Thus if {q " R : G1($, q, +) < *} #= ! then, by convexity andour assumption, G1($, q, +) has a minimum which must be greater than 0. There-fore, by the approximation property of pressure (see Proposition 2.3) we can findn " N such that the T -invariant compact set "n satisfies

P!n(q(") $#)) + log |T #|) > 0

for all q " R and

limq$%"

P!n(q(") $#)) + log |T #|) = limq$"

P((q(") $#)) + log |T #|) = *.

Moreover, the function q & P!n(q(")$#)) + log |T #|) is real analytic and strictlyconvex (see [SU, Sa2]). Thus, there exists a unique point qc " R such that

-

-qP!n(q, $, +)

///q=qc

= 0.

Denote by µc the unique equilibrium state for the potential qc(") $#)) + log |T #|restricted to "n. Note that such a measure exists because the space "n is compactand the potential Holder. Then

R'dµcR$dµc

= $ and so

h(µc)) +%(µc) > 0.

Since µc is ergodic we have that µc(J($)) = 1 and dimHµc = h(µc)((µc)

( +. Thiscompletes the proof. !

The following lemma now completes the proof of the lower bound for the casewhen $ " U .

Lemma 4.2. If $ " U then for all q " R and any & > 0 we have that

P (q(# ) $")) (+($)) &) log |T #|) > 0.

Proof. By the definition of +($) there exists a T -invariant measure µ such thatR'dµR$dµ

= $ and h(µ)((µ) > +($)) &. Thus, for any q " R, the variational principle yields

P (q(") $#)) (+($)) &) log |T #|) (

q

0+"dµ) $

+#dµ

1) (+($)) &)%(µ) + h(µ) =

)(+($)) &)%(µ) + h(µ) > 0.

!

For $ /" U we cannot use the argument in Lemma 4.2. Instead, we need to usea sequence of measures. Fix $ /" U and let

s = lim!$0

supµ!MT

'h(µ)%(µ)

:////

)"dµ)#dµ

) $

//// ' &

(.

Lemma 4.3. There exists a sequence of T -ergodic measures (µn)n such that,limn$"

h(µn)((µn) = s and limn$"

R'dµnR$dµn

= $.


Proof. It follows from the definition of s that we can find a sequence of invariantmeasures with this property. In order to construct a sequence of ergodic measureswith the desired property we proceed as follows. Let µ be an invariant measurewith max{h(µ),

)#dµ} < *. For any n " N we can define a Tn-ergodic Bernoulli

measure *n with *n([i1, . . . , in]) = µ([i1, . . . , in]) for each n " N. If we let ,n =1n

$n%1i=0 *n + T%1 then ,n is a T -ergodic measure. Moreover limn$"

h()n)(()n) = h(µ)

((µ)

and limn$"

R'd)nR$d)n

=R

'dµR$dµ

. !

Thus we have a sequence of T -invariant ergodic measures (µn)n such that

limn$"

h(µn)%(µn)

= s and limn$"

)"dµn)#dµn

= $.

If lim supn$")

#dµn < * then we can adapt the method in Lemma 3.2 to finda sequence of T -invariant measures (,n)n with limn$"

h()n)(()n) = s and

R'd)nR$d)n

=$. We can then proceed as in the case when $ " U . On the other hand, iflim supn$"

)#dµn = * we can adapt the technique in [IJ, Section 7] which is

in turn based on the method in Gelfert and Rams, [GR] to prove the followingproposition to complete the proof.

Proposition 4.4. If there exists a sequence of T -ergodic measures (µn)n suchthat limn$"

h(µn)((µn) = s,

R'dµnR$dµn

= $ and limn$")

#dµn = * then there exists a

measure ,, such that dimH, = s and limn$"Sn'(x)Sn$(x) = $ for ,-almost all x.

Note that the measure , is not required to be invariant and in certain cases aninvariant measure with the required property will not exist.

Proof of Proposition 4.4. Note that it su%ces to proof this in the case where$m > 0 since if this is not the case we can add a suitable constant multiple of # to" to ensure it is the case. We let

$n :=)

"dµn)#dµn

and sn =h(µn)%(µn)

.

We may also assume that the sequence sn is monotone increasing. Finally we fixa sequences of positive real numbers 0 < +n such that

8"n=1(1 ) +n) > 0 and fix

0 < &< infn!N min{$n, sn,)

#ndµn}.

Lemma 4.5. For each measure µn there exists a set Jn and jn " N such thatµn(Jn) > 1) +n and for all x = &(i) " Jn and j ( jn we have that

1. For all y " I(i1, . . . ij), Sj#(y) "9j9)

#dµn ) &/2n::

, j9)

#dµn + &/2n:

and Sj"(y)/Sj#(y) " ($n ) &/2n, $n + &/2n).2. |I(i1, . . . , ij)| " (j(%(µn) ) &/2n), j(%(µn) + &/2n)) and log(µn(I(i1,...,ij)))

log |I(i1,...,ij)| "((sn ) &/2n, sn + &/2n).

Proof. This is a straightforward consequence of the Birkho! ergodic theorem andthe Shannon-McMillan-Brieman Theorem combined with Egorov’s Theorem andour assumptions on ", #, log |T #|. !

We now define a sequence of natural numbers kn using the following inductiveprocedure. We let k1 satisfy the following conditions:

1. k1 ( j1,


2. &1k1

)#dµ1 ( 4j2$2

)#dµ2,

3. &1k1%(µ1) ( 4j2$2%(µ2),4. k1& ( 4j2%(µ2).

For n ( 1 we can choose kn+1 to satisfy:1. (kn+1 ) kn)&

)#dµn+1 (

//2n+2kn($n+1 ) $n))

#dµn

//,2. &kn+1

)#dµn+1 ( 2n+2jn+2$n+2

)#dµn+2,

3. (kn+1 ) kn)&%(µn+1) ( |2n+2kn(sn+1 ) sn)%(µn)|,4. &kn+1%(µn+1) ( 2n+2jn+2sn+2%(µn+2),5. (kn+1 ) kn)& ( 2n+2|%(µn+1)) %(µn)|6. kn+1& ( 2n+2jn+2%(µn+2).

Now consider a point x " J1!'%k1(J2)!'%k2(J3)! · · · . By the construction of thesets Jn and the values kn it follows that limn$"

Sn'(x)Sn$(x) = $. We can also define

a measure supported on this set as follows. Let *n denote the finitely supportedmeasure such that

*n(I(i1, . . . , ikn)) ='

0 if I(i1, . . . , ikn) ! Jn = !µn(I(i1, . . . , ikn)) if I(i1, . . . , ikn) ! Jn #= !.

It follows that *n(Jn) ( 1) +n and thus if we define the measure

* = *1 , *2 + T%k1 , *3 + T%k2 , · · ·

we will have

*(J1 ! '%k1(J2) ! '%k2(J3) ! · · · ) (";

n=1

(1) +n) > 0

and we can normalise to a measure ,. Let C = (8"

n=1(1) +n))%1

Lemma 4.6. For , almost all x we have that

limn$"

Sn"(x)Sn#(x)

= $ and lim infr$0

log ,(B(x, r))log r

( s.

Proof. By the construction of the measure , we have that for , almost all x = ((i),where i " #, is that

limn$"

Sn"(x)Sn#(x)

= $ and limn$"

log ,(I(i1, . . . , in))log |I(i1, . . . , in)| = s.

To complete the proof we will fix r > 0 su%ciently small. We fix n ( 2 and first ofall consider the case when

e%kn(((µn)%!/2n"1) ( r ( e%kn(((µn)%!/2n"1)%jn+1(((µn+1)+!/2n).

In this case B(x, r) contains at most ejn+1(((µn+1)+!/2n) sets of the form [i1, . . . , ikn ].Thus

log µ(B(x, r)) ' log C + kn%(µn)(sn ) &/2n%1) + jn+1(%(µn+1) + &/2n)' )kn%(µn)(sn ) &n) ' log C + (sn ) &n) log r.

Now consider the case where jn+1 ' k ' kn+1 ) kn and

e%kn(((µn)%k(((µn+1)%!/2n)) ( r > e%kn(((µn)%(k+1)(((µn+1%!/2n+1)).


Thus we have that

log µ(B(x, r)) ' log C + log %(µn+1) + log µ(I(i1, . . . , ikn+k))' log C log %(µn+1) + kn%(µn)(sn ) &/2n%1) + k%(µn+1)(sn+1 ) &/2n%1)' log C + (sn ) &/2n%1) log r.

Finally note that by the definition of kn

e%kn(((µn)%kn+1)%kn(((µn+1)%!n+1/2) ' e%kn(((µn)%!)n

and so we have considered all possibles case for r < e%k2(((µ1)%!1) and the resultfollows. !

The proof of Proposition 4.4 is now complete.

5. Proof of Theorem 1.4

We now turn to the question of when the spectrum is analytic. Throughout thissection ", # are both locally Holder, $M < * , "/# is uniformly bounded aboveand

limx$0

#(x)log |T #(x)| = *.

An important consequence of our assumptions is that if µ is a T -invariant measurefor which

R'dµR$dµ

" U then by Lemma 3.1)

#dµ cannot be too large and thus %(µ)cannot be too large.

Lemma 5.1. For any C1 > 0 there exists C2 > 0 such that if µ is a T)invariantmeasure for which %(µ) ( C2 then

)#dµ ( C1%(µ).

Proof. Let + > 0, then there exists A " (0, 1) such that if x " (0, A) then

log |T #(x)|#(x)

< +.

Moreover, there exists a constant C > 0 such that if x " (A, 1) then log |T #(x)| < C.Let µ be a T)invariant measure satisfying %(µ) ( C, then we have that

%(µ) =+ 1

0log |T #(x)|dµ =

+ A

0log |T #(x)|dµ +

+ 1

Alog |T #(x)|dµ

'+ A

0log |T #(x)|dµ + µ([A, 1])C '

+ A

0log |T #(x)|dµ + C.

That is

%(µ)) C '+ A

0log |T #(x)|dµ.

We thus have,

+ () A0 log |T #(x)|dµ

) A0 #dµ

( %(µ)) C) A0 #dµ

.

Therefore, since # > 0, we have

(4)+ 1

0#dµ (

+ A

0#dµ ( %(µ)) C

+.


Thus, given C1 > 0 choose + > 0 such that 1) +C1 > 0 and let

C2 >C

1) +C1.

Therefore, if %(µ) > C2 then %(µ)(1) +C1) > C, which implies that%(µ)) C

+> C1%(µ).

Combining this with equation (4) we obtain that+ 1

0#dµ (

+ A

0#dµ ( %(µ)) C

+( C1%(µ),

which completes the proof. !Lemma 5.2. For $ " U we have that for any + > 0 either

1. G1($, q, +) < * for all q > 0 or2. G1($, q, +) < * for all q < 0.

Proof. Since $ " U we know that either $ > $ or $ < $. To start we will assumethat $ > $. We fix ) " ($,$) and q > 0. By the variational principle we need toshow that there is a uniform upper bound on

h(µ) + q

0+"dµ) $

+#dµ

1) +%(µ)

for all T -invariant probability measures µ. We will split the set of invariant measuresinto two sets depending on whether or not

)"dµ)#dµ

' ).

Firstly, if)

"dµ ' ))

#dµ then q()

"dµ ) $)

#dµ) ' q() ) $))

#dµ < 0.Furthermore, by taking C1 = ()q() ) $))%1 in Lemma 5.1 it follows that thereexists C2 > 0 such that if %(µ) ( C2 we have that (using Ruelle’s inequality)

+#dµ ( C1%(µ) ( C1h(µ) =

1)q() ) $)

h(µ).

That is,

)q() ) $)+

#dµ ( h(µ).

We therefore have

h(µ) + q

0+"dµ) $

+#dµ

1) +%(µ) '

)q() ) $)+

#dµ + q() ) $)+

#dµ) +%(µ) = )+%(µ) ' 0.

On the other hand, if )"dµ)#dµ

( ).

By Lemma 3.1 there exists a constant K # such that)

#dµ < K #. We also have thatthere exists a uniform upper bound for K ## > 0 for

%(µ))#dµ

< K ##.


Therefore, there exists K > 0 such that max<h(µ), %(µ),

)#dµ

=' K. This means

that

h(µ) + q

0+"dµ) $

+#dµ

1) +($)%(µ) ' K + q$MK.

So for $ " ($,$M ) we have that for all q > 0 the function G1($, q, +) is boundedabove,

G1($, q, +) < *.

For the case where $ " ($m, $) we fix q < 0 and ) " ($,$). The same argumentallow us to conclude that G1($, q, +) < *.

!

As a result of Lemma 5.2 we can investigate the behaviour of the function G1

when + = +($) and $ " U .

Lemma 5.3. For $ " U we have that G1($, q, +($)) ( 0 for all q " R and thateither

1. there exists a unique qc #= 0 such that G1($, q, +($)) = 0 and

-

-qG1(q, +($), $)

//q=qc

= 0

or2. +($) = dimH" and P ()+($) log |T #|) = 0.

Proof. We first prove that for all q " R we have G1($, q, +($)) ( 0. Note that since$ " U , there exists a constant C > 0 and a sequence of T -invariant measures (µn)n

such that %(µn) ' C,R

'dµnR$dµn

= $ and limn$"h(µn)((µn) = +($). We thus have for all

q " R and n " N that

G1($, q, +($)) ( h(µn)) +($)%(µn).

Letting n tend to infinity we obtain

G1($, q, +($)) ( 0.

If +($) = dimH" the above argument together with Bowen’s formula,

P ()dimH" log |T #|) ' 0,

implies that P ()+($) log |T #|) = 0.From now on we can suppose that +($) < dimH". In particular,

0 < P ()+($) log |T #|),

note that it could be infinite. Moreover, since $m < $ < $M we have that

limq$"

G1($, q, +($)) = limq$%"

G1($, q, +($)) = *.

Now, in order to obtain a contradiction, suppose there exists C1 > 0 such that

G1($, q, +($)) ( C1 > 0

for all q " R. Then we can find a compact T -invariant subset "n, where "n $ ",such that for every q " R we have

0 < C1/2 ' PT |!n(q(") $#)) +($) log |T #|) < *


and such that the following holds

limq$%"

PT |!n(q(") $#)) +($) log |T #|) = * and

limq$"

PT |!n(q(") $#)) +($) log |T #|) = *.

Thus there will exist a turning point qt " R and the equilibrium state µt associatedto qt(")$#))+($) log |T #| restricted to "n satisfying

R'dµtR$dµt

= $ and h(µt)((µt)

> +($),which is a contradiction.

Thus, there exists a nonzero point qc " R such that G1($, qc, +($)) = 0. SinceG1($, 0, +(P ()+($) log |T #|) > 0 and

limq$"

G1($, q, +($)) = limq$%"

G1($, q, +($)) = *

it follows from Lemma 5.2 that qc must be a turning point and the result follows.!

We can now split the region ($m, $M ) up into three intervals depending onwhether $ in U or not and if so which case of Lemma 5.3 is true.

Lemma 5.4. We can write ($m, $M ) = J1 % J2 % J3 where J1, J3 are intervals orthe empty set, J2 is an interval or a single point such that E = [$,$] $ J2 and

1. For $ " J2, dimHJ($) = dimH".2. For $ " J1%J3 we have that there exists a unique qc #= 0 such that P (qc(")

$#)) +($) log |T #| = 0 and ++q G1(qc, +($), $) = 0

Proof. We define

J2 = {$ " ($m, $M ) : P (q(") $#)) (dimH") &) log |T #|) ( 0 for q " R and & > 0}and also let

J1 = {$ " ($m, $M ) : $ < ) for every ) " J2} andJ3 = {$ " ($m, $M ) : $ > ) for every ) " J2}.

For $ " J2 we have combining Lemma 4.1 with Theorem 1.3 that dimHJ($) =dimH".

If $1, $2 " J2 and ) " ($1 $2) then ) " J2 by the convexity of the pressurefunction. Therefore J2 is either a single point or an interval and it thus follows thatboth J1 and J3 are either empty or intervals.

To show that E = [$,$] $ J2 we fix $ " E and choose & > 0 and ) satisfyingthat there exist a T -invariant measure µ such that %(µ) < *, h(µ)

((µ) > dimH") &/2

andR

'dµR$dµ

= ). Suppose that ) > $ and let $ " ($, )). Since $ " E we can find asequence of T -invariant measures ,n and 0 < pn < 1 such that

limn$"

)"d,n)#d,n

= $, limn$"

pn%(,n) = 0 and limn$"

pn

+#d,n = *.

Thus, if we consider the measures *n = pn,n +(1)pn)+, then limn$"

R'd"nR$d"n

= $

and lim supn$"h("n)(("n) ( dimH") &/2. Therefore, by taking an appropriate convex

combination, we can find a T -invariant measure , such thath(,)%(,)

> dimH") & and)

"d,)#d,

= $.


By the variational principle, this means that for all q " R we have

P (q(") $)) (dimH") &) log |T #|) > 0.

The case when $ > ) can be dealt with analogously by using the interval (),$).Finally note that if ) = $ = $ then we can verify that ) " J2 directly by thevariational principle.

Now fix $ " J1. This means that $ " U and that there exists q " R such that

P (q(") $#)) (dimH") log |T #|) < 0.

Thus, by Lemma 5.3 we know that +($) < dimH" and since then

P ()+($) log |T #(x)|) > 0

we must be in case 1 of Lemma 5.3. The case when $ " J3 can be dealt withanalogously. This completes the proof. !

The three intervals J1, J2, J3 will be exactly the three intervals in the statementof Theorem 1.4. To complete the proof of Theorem 1.4 we simply need to provethat $ & dimHJ($) varies analytically on J1 and J3.

Lemma 5.5. The function $ & J($) is analytic in J1 and J3.

Proof. We will proof this result for the interval J1, since the proof for the intervalJ3 is analogous. Note that for $ " J1 we have that dimHJ($) = +($). We will letG2 : J2 - R+ - (0,dimH") & R be defined by

G2($, q, +) =-

-qG1($, q, +)

and note that since G1 is finite throughout the specified region G2 is well defined.Let G : J2 - R+ - (0,dimH") & R2 be defined by

G($, q, +) = (G1($, q, +), G2($, q, +)).

For each $ " J1 by Lemma 5.4 that there exists a unique q($) " R such that

G($, q($), +($)) = (0, 0) and dimHJ($) = +($).

Note that G is finite and varies analytically in each of the three variables q, $, +throughout its range. Thus, to complete the proof we wish to apply the ImplicitFunction Theorem. To be able to do this it su%ces to show that the matrix

-+G1+&

+G1+q

+G2+&

+G2+q

.

is invertible at each point ($, q($), +($)). At such points +G1+q = 0 and +G1

+& < 0(Indeed, it corresponds to the Lyapunov exponent with a minus sign). So we needto show that +G2

+q is nonzero at ($, q($), +($)). If " ) $# is not cohomologousto a constant then the function G1 is strictly convex in the variable q and theproof is complete. To deduce that ") $# is not cohomologous to a constant notethat $m < $ < $M and thus there exist T -invariant measures µ1 and µ2 such that)

(")$#)dµ1 < 0 and)

(")$#)dµ2 > 0 therefore ")$# cannot be cohomologousto a constant. !


6. Discontinuities in the spectrum and applications to non-uniformlyhyperbolic systems

In this section we show that in the setting of Theorem 1.4 it is possible that thefunction $ & dimHJ($) is discontinuous at a point in ($m, $M ). We stress thatthis is a new phenomenon that does not occur in the uniformly hyperbolic settingwith regular potentials. Here we not only establish conditions for this to happen,but also exhibit very natural non-uniformly hyperbolic dynamical systems wherethese conditions are satisfied and therefore, regular potentials have discontinuousspectrum.

We will denote by µSRB the T-invariant measure of maximal dimension and bys" := inf{s : P ()s log |T #|) < *}. We have the following result,

Proposition 6.1. Let " " R and # " R" be such that

limx$0

#(x)log |T #(x)| = *

Assume that1. limx$0

'(x)$(x) = 0

2. there exists a T -invariant probability measure, µ, such that)

"dµ < 0;3. dimH" > s",4.

)"dµSRB > 0

then the function $ & dimHJ($) is discontinuous at $ = 0.

Proof. To start we let $' =R

'dµSRBR$dµSRB

and note that we have under the notation ofTheorem 1.4 that J1 = [$m, 0], J2 = [0, $'] and J3 = [$', $M ]. It also follows fromTheorem 1.4 that dimHJ(0) = dimH". Assume by way of contradiction that thefunction $ .& dimH(J($)) is continuous at $ = 0. In particular it is continuos fromthe left. That is

lim#$0"

dimHJ($) = dimH".

There exist a sequence ($n)n such that for every n " N we have $n " ($m, 0) and

limn$"

dimHJ($n) = dimH".

In virtue of the variational principle there exists a sequence of T)invariant measures(µn)n such that

0)"dµn)#dµn

) $n

1' 1

nand

0h(µn)%(µn)

) dimHJ($n)1' 1

n

In particular, for each n " N we have)

"dµn < 0, limn$"

R'dµnR$dµn

= 0 and

limn$"h(µn)((µn) = dimH". The existence of such sequence imply by [FJLR, Propo-

sition 6.1] 1 that there exists a T -invariant measure , such that)

"d, ' 0 andh())(()) = dimH". Thus , is an equilibrium state for the potential )(dimH") log |T #|and since this equilibrium state is unique we must have , = µSRB. However,

1The method of proof of [FJLR, Proposition 6.1] allows us to deduce the existence of such ameasure !. Note that in the notation of [FJLR, Proposition 6.1] we take "i = " for all i ! Nand note that while "i is required to be bounded for [FJLR, Proposition 6.1] to hold, it can beadapted to this setting by replacing

R"id! = #i by

R"dµ " lim sup

R"idµn " 0 by using [JMU,

Lemma 1].


)"dµSRB > 0 and

)"d, ' 0 which is obviously a contradiction. Therefore

there exists no such sequence of measures and $ & dimHJ($) is discontinuousat $ = 0. !

6.1. Manneville-Pomeau. In this subsection we discuss an example that illus-trates how our results Theorem 1.4 and Proposition 6.1 can be applied to certainclasses of non-uniformly expanding maps. In [JJOP] the authors proved a vari-ational principle for Birkho! averages for continuous potentials and certain non-uniformly expanding maps. A particular case of this is the Manneville-Pomeaumap, F : [0, 1] .& [0, 1], which is the map defined by F (x) = x + x1+, mod 1,where 0 < . < 1. This map has an indi!erent fixed point at x = 0 and it hasan absolutely continuous (with respect to Lebesgue) invariant probability measure.We will denote A = %"n=0F

%n({0}) and let 0 < t < 1 satisfy t+t1+, = 1. We definea partition P1 = {[0, t], [t, 1]} and Pn =

>n%10 T%nP1. For a function f : [0, 1] & R

letvarn(f) = sup

P!Pn

sup{|f(x)) f(y)| : x, y " P}.

We will assume that varn(f) = O(/n) for some 0 < / < 1, f(0) = 0, that f isnon-negative in a neighbourhood of 0, let

$m = inf'+

fdµ : µ is F -invariant(

and

$M = sup'+

fdµ : µ is F -invariant(

.

We define,

J($) =

#x " [0, 1] : lim

n$"

1n

n%1,

i=0

f(F ix) = $

%.

We stress that in [JJOP] the Hausdor! dimension of the level sets, dimHJ($), isfound for a more general class of functions. The result is that if $ " [$m, $M ] \ 0then

dimHJ($) = sup'

h(µ)%(µ)

: µ "MF ,

+f dµ = $

(

and it is also shown that dimHJ(0) = 1.With these stronger assumptions on our function f we can use our results from

Theorem 1.4 to say more about the function $ & dimHJ($). It is well known thatF can be related to a countable EMR map T .

n(x) ='

1 if x " [t, 1]inf{n : Fn(x) " I}+ 1 if x /" [t, 1]

and T (x) = Fn(x)(x). Note that T is an EMR map and we have that " = [0, 1]\Aand since A is a countable set dimH" = 1. We can also calculate s" = ,

,+1 (indeed

see the proof of [Sa2, Proposition 1]). We can define "(x) =$n(x)%1

i=0 f(F ix) and#(x) = rn(x) and note that "(x) " R and #(x) " R1. For $ " R let

X# ='

x " [0, 1]\A : limn$"

Sn"(x)Sn#(x)

= $

(.

Proposition 6.2. dimHJ($) = dimHX#.


Proof. It is immediate that J($) / X# for all $ except for $ = 0. We also have thatJ(0) $ X0%A and since A is a countable set it follows that dimHX0 ( dimHJ(0) =1. Thus dimHX0 = dimHJ(0) = 1.

For the case when $ #= 0 we first note that this implies that $ " U . Thus thereexists C > 0 such that for all x " X# we have that lim supn$"

Sn$(x)n < C. We

now let x " X# and note that limn$"Sn+1$(x)Sn$(x) = 1. Thus, if we let rn = Sn#(x)

we have that for rn ' k ' rn+1$k%1

i=0 f(F ix)k

'$rn+1

i=0 f(F ix)rn+1

rn+1

rn

and by taking limits as n & * we can see that x " J($) and so the proof iscomplete. !

This means that we can apply Theorem 1.4 to obtain more information aboutthe function $ & dimHJ($).

Theorem 6.3. If $m ' 0 = f(0) ' $' =)

fdµSRB < $M then1. The function $ & dimHJ($) is analytic on ($m, 0) and ($', $M ).2. dimHJ($) = 1 for $ " [0, $']3. If $m < 0 then $ & dimHJ($) is discontinuous at $ = 0.

Proof. To prove this Theorem we first note that the potentials ", # satisfy theassumptions for Theorem 1.4. Note that the absolutely continuous measure for Tprojects to the absolutely continuous measure for F . Thus, in Theorem 1.4 wehave that J3 = ($', $M ). We can determine that if limn$"

Sn"(x)n = * then

limn$"Sn'(x)Sn$(x) = 0 and so U = 0. Thus J2 = (0, $') and we can conclude that

J3 = ($m, 0). The first two parts of the Theorem now immediately follow fromTheorem 1.4 and the final part follows since s" < 1 and thus the assumptions forProposition 6.1 are met. !Remark 6.4. We would be able to proof analogous results if $' < 0 and f isnegative in a neighbourhood of 0. The theorem would hold with weaker assumptionon the function f . What we need is that " is locally Holder.

7. Multifractal Analysis for suspension flows

Let T be an EMR map and 0 : (0, 1] & R a positive function in R". We considerthe space

Y := {(x, t) " (0, 1]- R : 0 ' t ' 0(x)},with the points (x, 0(x)) and (T (x), 0) identified for each x " (0, 1]. The suspensionsemi-flow over T with roof function 0 is the semi-flow $ = (1t)t&0 on Y defined by

1t(x, s) = (x, s + t) whenever s + t " [0, 0(x)].

In particular,1-(x)(x, 0) = (T (x), 0).

Because of the Markov structure of T the flow $ can be coded with a suspensionsemi-flow over a Markov shift defined on a countable alphabet. Let g : Y & R bea potential and define

K($) :='

(x, r) " Y : limt$"

1t

+ t

0g(1s(x, r))ds = $

(.


We define the Birkho! spectrum of g by

B($) := dimH(K($)).

It turns out that the results obtained to study multifractal analysis for quotientsallow us to study the Birkho! spectrum for flows.

Remark 7.1 (Invariant measures). We denote by M# the space of $-invariantprobability measures on Y . Recall that a measure µ on Y is $-invariant if µ(1%1

t A) =µ(A) for every t ( 0 and every measurable set A $ Y . Consider as well the spaceM(T ) of T -invariant probability measures on (0, 1] and

M(T )(0) :='

µ "M(T ) :+

0dµ < *(

.

Denote by m the one dimensional Lebesgue measure and let µ "M(T )(0) then itfollows directly from classical results by Ambrose and Kakutani [AK] that

(µ-m)|Y /(µ-m)(Y ) "M#.

Moreover, If 0 : (0, 1] & R is bounded away from zero then there is a bijectionbetween the spaces M# and M(T )(0).

Remark 7.2 (Kac’s formula). Given a continuous function g : Y & R we definethe function 'g : (0, 1] & R by

'g(x) =+ -(x)

0g(x, t) dt.

The function 'g is also continuous, moreover

(5)+

Yg dR(,) =

)$ 'g d,)$ 0 d,

.

Remark 7.3 (Abramov’s formula). The entropy of a flow with respect to aninvariant measure can be defined by the entropy of the corresponding time onemap. Abramov [A] and later Savchenko [Sav] proved that if µ " M# is such thatµ = (, -m)|Y /(, -m)(Y ), where , "M(T ) then

(6) h#(µ) =h%(,))

0d,.

The following Lemma establishes a relation between the level sets determinedby Birkho! averages for the flow and the level sets of quotients for the map T .

Lemma 7.4. If

limt$"

1t

+ t

0g(1s(x, r))ds = $,

then

limn$"

$ni=0 'g(T ix)$ni=0 0(T ix)

= $.

Proof. Denote by

0m(x) :=m%1,

i=0

0(T ix).


We have that+ -m(x)

0g(1s(x, r)) ds =

m%1,

i=0

+ -i+1(x)

-i(x)g(1s(x, r) ds =

m%1,

i=0

+ -(T ix)

0g(1s(x, r) ds =

m%1,

i=0

'g(T ix).

In particular if

limt$"

1t

+ t

0g(1s(x, r))ds = $,

since t &* implies that m &*, we have that

limm$"

10m(x)

m%1,

i=0

'g(T i(x)) = limm$"

$m%1i=0 'g(T i(x))

$m%1i=0 0(T i(x))

= $.

!Let

J($) :='

x " (0, 1] : limn$"

$ni=0 'g(T ix)$ni=0 0(T ix)

= $

(.

Our main results establishes that we can compute the Hausdor! dimension ofK($) once we know the Hausdor! dimension of J($).

Theorem 7.5. Let $ " R be such that K($) #= !, then

dimHK($) = dimHJ($) + 1.

We divide the proof of this results in a couple of Lemmas. The proof of the upperbound for the dimension of K($) in terms of he dimension of J($) is simpler.

Lemma 7.6. Let $ " R be such that K($) #= !, then

dimHK($) ' dimH (J($)- R) = dimHJ($) + 1.

Proof. First note that if (x, r) " K($) then by virtue of Lemma 7.4 we have thatx " J($). Also if (x, r) " K($) then (x, s) " K($) for every s " [0, 0(x)). Wetherefore have

K($) $<(x, r) " R2 : x " J($) and r " [0, 0(x)]

=.

The box dimension of R and its Hausdor! dimension coincide, both are equal toone. Therefore, the dimension of the Cartesian product is the sum of the dimensionsof each of the factors (see [Fa, p.94]). The result now follows. !

In order to prove the lower bound we will use an approximation argument.

Remark 7.7 (Compact setting). Let C $ Y is a compact $)invariant set andconsider the restriction of g to the set C (which is a Holder map). Then, it wasproven by Barreira and Saussol [BS2, Proposition 6] that, (x, r) " K($) if and onlyif x " J($).

The following Lemma completes the proof of Theorem 7.5.

Lemma 7.8. Let $ " R be such that K($) #= !, then

dimHK($) ( dimH (J($)- R) = dimHJ($) + 1.


Proof. Let (Cn)n be an increasing sequence of a compact $)invariant set thatexhaust Y , denote by Cn the projection of Cn onto (0, 1]. We define

Kn($) := K($) ! Cn and Jn($) := J($) ! Cn.

By Remark 7.7 we have that

Kn($) =<(x, r) " R2 : x " Jn($) and r " [0, 0(x)]

=.

Hence, using the formula for the Hausdor! dimension of a Cartesian product (see[Fa, p.94]), we have

dimHKn($) = dimHJn($) + 1.

Therefore

limn$"

dimHKn($) = limn$"

(dimHJn($) + 1) = dimHJ($) + 1 ' dimHK($).

!

It is a direct consequence of Theorem 7.5 that in order to describe the behaviourof the function B($) we only need to understand b($). Recall that " denotes therepeller for T . The following is a version of Theorem 1.4 in the suspension flowsetting. It thus, describe the regularity properties of the map B($).

Theorem 7.9. Let T be an EMR map, 0 " R" a roof a function and $ theassociated suspension semi-flow. Let g : Y & R be a continuous potential such that'g " R. Assume that

limx$0

0(x)log |T #(x)| = *,

then there exist three pairwise disjoint intervals J1, J2, J3 such that1. The domain of K($) can be written as J1 % J2 % J3,2. J1 ' J2 ' J3

3. The function $ & dimHK# is analytic on J1 and J3

4. For $ " J2, dimHK# = dimH" +15. It is possible that J1 = !, J3 = ! or that J2 is a single point.

8. Continued fractions

In this section we consider examples involving the Gauss map and, hence, thecontinued fraction expansion of a number. Every irrational number x " [0, 1] canbe written in a unique way as a continued fraction,

x =1

a1 +1

a2 +1

a3 + . . .

= [a1a2a3 . . . ],

where ai " N. It well known that the Gauss map, G : (0, 1] & (0, 1], acts as theshift in the continued fraction expansion (see [EW, Chapter 3]), that is

G([a1a2a3 . . . ]) = [a2a3a4 . . . ].


8.1. Arithmetic and Geometric averages. Let # : [0, 1] & R be defined by#([a1a2a3 . . . ]) := a1. Note that the Birkho! average of G with respect to thepotential # is nothing but the arithmetic average of the digits in the continuedfraction expansion

limn$"

1n

",

n=1

#(Gn(x)) = limn$"

a1 + a2 + · · ·+ an

n.

Note that [EW, p.83] for Lebesgue almost every point the arithmetic average isinfinite. The level sets induced by the arithmetic averages where studied in [IJ].Consider now the function " : [0, 1] & R defined by "(x) = log a1. The Birkho!average of G with respect to that function is the logarithm of the geometric average,

limn$"

1n

",

n=1

"(Gn(x)) = limn$"

log n0

a1a2 · · · an.

For Lebesgue almost every point this sum takes the value [EW, p.83]

log

- ";

n=1

0(n + 1)2

n(n + 2)

1log n/ log 2.

.

The level sets determined by the geometric average were studied in [FLWW, KS,PW]. On the other hand, the Birkho! average corresponding to log |G#(x)| is theLyapunov exponent of the point x, that is

%(x) := limn$"

1n

",

n=1

log |G#(Gn(x))|

if this limit exists. This number measures the exponential speed of approximationof an irrational number by its approximants, which are defined by,

pn

qn:= [a1 . . . an].

That is (see [PW]) ////x)pn

qn

//// 1 exp()n%(x)).

For Lebesgue almost every point this number equal to [EW, p.83],

(2

6 log 2.

The multifractal analysis for this function has been studied in [PW] and [KS]. Thetechniques developed in this paper allow us to study the following related level setsof the form

J($) :='

x " [0, 1] : limn$"

log(a1a2 · · · an)a1 + a2 + · · ·+ an

= $

(.

Note that the quotient defining the level set is the quotient of the logarithm of thegeometric average with the arithmetic average. Indeed,

log(a1a2 · · · an)a1 + a2 + · · ·+ an

=1n1n

log(a1a2 · · · an)a1 + a2 + · · ·+ an

=log n

0a1a2 · · · an

a1+a2+···+ann

.

Lemma 8.1. We have that $m = 0 and $M = log 33 . Moreover, the set J(0) has

full Lebesgue measure.


Proof. We have that $m = 0. This is clear, just consider the number x = [11111 . . . ].It is easy to construct numbers belonging to J(0) as the following example shows,let x = [1, 2, 22, 23, . . . , 2n . . . ]. That is the number for which the digits in thecontinued fraction expansion are in geometric progression with ratio equal to 2.Then

a1 + a2 + · · ·+ an = 1 + 2 + 22 + 23 + · · ·+ 2n%1 = 2n ) 1.

On the other hand

log(a1a2 · · · an) = log(12223 · · · 2n%1) = log 21+2+3+···+(n%1) =(n) 1)n

2log 2.

Therefore

limn$"

log(a1a2 · · · an)a1 + a2 + · · ·+ an

= limn$"

(n%1)n2 log 22n ) 1

= 0.

The fact that the set J(0) has full Lebesgue measure is a direct consequence of thefact that for Lebesgue almost every point the arithmetic average is infinite and thegeometric one is finite.

On the other hand we have that $M = (log 3)/3. Indeed, it is well known thatthe arithmetic average is larger than the geometric one

a1 + a2 + · · ·+ an

n( n0

a1a2 · · · an,

with equality if and only if a1 = a2 = a3 = · · · = an. Therefore the maximum ofthe quotient

log(a1a2 · · · an)a1 + a2 + · · ·+ an

is achieved in an algebraic number of the form x = [a, a, a, . . . ]. In this case weobtain

log(a1a2 · · · an)a1 + a2 + · · ·+ an

=log a

a.

The maximum of the function f(x) = (log x)/x is attained at x = e. Since a " Nwe have that the maximum is (log 3)/3. !

Lemma 8.2. We have that $ = $ = 0. In particular U = [$m, $M ] \ ($,$) =(0, (log 3)/3)

Proof. Note that

limx$0

"(x)#(x)

= limn$"

log n

n= 0.

That is $ = $ = 0. Therefore U = ($m, $M ) = (0, (log 3)/3). !

Therefore, a direct consequence of Theorem 1.3 is that

Proposition 8.3. For every $ " (0, (log 3)/3) we have that

b($) = dimHJ($) = supµ!M(G)

'h(µ)%(µ)

:)

"dµ)#dµ

= $

(.

Lemma 8.4. We have that

limx$0

#(x)log |T #(x)| = *.


Proof. Note that if x " (1/(n + 1), 1/n) then #(x) = n and 2 log n ( log |T #(x)|.Thus,

limx$0

#(x)log |T #(x)| ( lim

n$"

n

2 log n= *.

!

In particular we have proved that the assumptions of Theorem 1.4 are satisfied.

8.2. Weighted arithmetic averages. Let 2, ) > 0 consider the level sets definedas the quotient of weighted arithmetic averages,

J($) :='

x " [0, 1] : limn$"

a.1 + a.

2 + · · ·+ a.n

a/1 + a/

2 + · · ·+ a/n

= $

(.

Let # : [0, 1] & R be defined by #([a1a2a3 . . . ]) := a/1. Note that the Birkho!

average of G with respect to the potential # is nothing but the weighted arithmeticaverage of the digits in the continued fraction expansion

limn$"

1n

",

n=1

#(Gn(x)) = limn$"

a/1 + a/

2 + · · ·+ a/n

n.

In an analogous way we define " : [0, 1] & R by #([a1a2a3 . . . ]) := a.1 . The

level sets determined by Birkho! averages of the potential # where studied in [IJ,Proposition 6.3]. In that context there exists essentially two di!erent types ofbehaviour, depending if 2 " (0, 1) or if 2 ( 1.

Remark 8.5. If ) > 2 then

$ = $ = limx$0

"(x)#(x)

= limn$"

n/%. = 0.

Since the level sets are defined by the quotient of positive numbers we have that$m = 0. Remark that if x = [a, a, a, . . . ] then

limn$"

a. + a. + · · ·+ a.

a/ + a/ + · · ·+ a/= a/%. .

In particular if x = [1, 1, 1, . . . ] we have that $ = 1 " [$m, $m]. On the other handnote that if a " N then a/ ' a. , therefore

limn$"

a. + a. + · · ·+ a.

a/ + a/ + · · ·+ a/' 1.

Thus, $M = 1.

Proposition 8.6. If ) > 2 then for every $ " (0, 1) we have that

b($) := dimHJ($) = supµ!M(T )

'h(µ)%(µ)

:)

"dµ)#dµ

= $

(

Proof. In Remark 8.5 we proved that $m = 0 and E = {0}. The result now followsby applying Theorem 1.3. !

Lemma 8.7. We have that

limx$0

#(x)log |T #(x)| = *.


Proof. Note that if x " [1/(n + 1), 1/n] then #(x) = n. and 2 log n ( log |T #(x)|.Thus,

limx$0

#(x)log |T #(x)| ( lim

n$"

n.

2 log n= *.

!

In particular we have proved that the assumptions of Theorem 1.4 are satisfied.

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Facultad de Matematicas, Pontificia Universidad Catolica de Chile (PUC), AvenidaVicuna Mackenna 4860, Santiago, Chile

E-mail address: [email protected]

URL: http://www.mat.puc.cl/~giommi/

The School of Mathematics, The University of Bristol, University Walk, Clifton,Bristol, BS8 1TW, UK

E-mail address: [email protected]

URL: http://www.maths.bris.ac.uk/~matmj

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