Multifractal Analysis of Weak Gibbs Measure

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Ergod. Th. & Dynam. Sys. (2003), 23, 17511784 c 2003 Cambridge University PressDOI: 10.1017/S0143385703000051 Printed in the United Kingdom

Multifractal analysis of weak Gibbs measures

and phase transitionapplication to some

Bernoulli convolutions

DE-JUN FENG and ERIC OLIVIER

Department of Mathematical Sciences, Tsinghua University, Beijing 100084,

Peoples Republic of China

(e-mail: [email protected]) Department of Mathematics, The Chinese University of Hong Kong, Shatin,

New Territories, Hong Kong

(Received6 June 2001 and accepted in revised form 23 January 2003)

Abstract. For a given expanding d-fold covering transformation of the one-dimensional

torus, the notion of weak Gibbs measure is defined by a natural generalization of the

classical Gibbs property. For these measures, we prove that the singularity spectrum

and the Lq -spectrum form a Legendre transform pair. The main difficulty comes from

the possible existence of first-order phase transition points, that is, points where the

Lq -spectrum is not differentiable. We give examples of weak Gibbs measure with phase

transition, including the so-called Erdos measure.

0. Introduction

The one-dimensional torus S1 := R/Z is endowed with the natural metric and dimH Mdenotes the Hausdorff dimension of any M S1 (by convention, dimH = ).Let Br (x) be the closed ball of radius r > 0 centered at x S1; the local dimensionof a Borel probability measure at x is by definition

DI M(x) := limr0

log (Br (x))

log r , (1)

provided that the limit exists. The level setE() ( R) associated to is the set of pointsx S1 such that DI M(x) exists and is equal to . The map dimH E() is called thesingularity spectrum of . Heuristic arguments using techniques of statistical mechanics

(see [16] for example) show that the singularity spectrum should be finite on a compact

interval denoted DO M() and is expected to be the Legendre transform conjugate of the

Lq -spectrum associated to (see Definition 1.3); that is, for all DO M(),dimH E() = inf{q (q); q R} =: (). (2)

http://www.paper.edu.cn


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1752 D.-J. Feng and E. Olivier

The multifractal analysis of a probability measure is concerned with rigorousarguments

ensuring that the Legendre transform formula (2) holds; the first multifractal formalismswere established for Gibbs [7, 36, 39], quasi-Bernoulli [4, 17] and self-similar measures

[6, 25, 33]: we refer to [34] for a general overview of the subject and complete references.

In this paper, we focus our attention on the notion of weak Gibbs measure defined in [46]

by a natural generalization of the classical Gibbs property (see Definition 1.1). Of special

interest are the g-measures, as considered in [22], which turn out to be weak Gibbs, as well

as a large class of the conformal measures studied in [19]. It follows from the variational

characterization of the g-measure proved in [26] that a non-ergodic g-measure cannot be

Gibbs. The existence of non-ergodic g-measures established in [3] shows that a weak

Gibbs measure need not be Gibbs.

Let T be an expanding d-fold covering transformation of S1 (with d 2) so that there

exists a Markov partition M1 ofT, byd

(semi-open) intervals; ifMn (n 1) denotes thepartition of S1 by the n-step basic intervals then M := n Mn is called the Markoviannet of T. From now on, we assume that is a weak Gibbs measure of a potential

defined on the symbolic space which codes the Markovian dynamics of T. Our approach

to the multifractal analysis of is classical; we consider the level sets E(|M) associatedto the Markovian local dimension DI M(|M) and we give (Theorem A) an intermediateLegendre transform formula, say, for any in the interior of DO M(),

dimH E(|M) = (), (3)

where the concave map is implicitly defined by a pressure equation. An important

and non-trivial step is to prove (Theorem B) that dim H E(|M) = dimH E(), when is a weak Gibbs measure. The last step is achieved by proving (Theorem C) that

coincides with the Lq -spectrum and we conclude (Theorem A) that the Legendre

transform formula (2) holds, when is weak Gibbs.

These generalizations are relevant essentially because the Lq -spectrum need not be

real-analytic/differentiable when is weak Gibbs. For the thermodynamic formalism on

lattices, a system is said to exhibit a phase transition when a thermodynamic function

displays a defect of analyticity at some critical value (see [ 40, ch. 5]). We shall say that the

real number qc is a phase transition point(respectively a first-order phase transition point)

if is not real-analytic (respectively not differentiable) at qc: this will make sense, for we

shall prove (Theorem C) that coincides with a thermodynamic function determined by

a pressure equation. Let us denote by (q+c ) (respectively (qc )) the right (respectively

left) derivative of at qc; if (q+c ) <

(qc ) then the mass distribution principle does

not apply to give the desired lower bound of dimH E(|M) when (q+c ) < < (qc ).Our argument depends on the tangency property of the topological pressure, which yields a

thermodynamic characterization of(q+) and (q) for any q R (Lemma 3.2) and ona formula which gives the Billingsley dimension of the generic points of a (not necessarily

ergodic) shift-invariant measure.

In 1 we describe the framework of the expandingd-fold covering transformation of the

one-dimensional torus and the thermodynamicformalism of the equilibrium state. Then we

give a definition of the weak Gibbs measure (Definition 1.1) making possible a rigorous

statement of Theorems A, B, C and A; the proofs of these theorems are given in 3.



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Multifractal analysis of weak Gibbs measures 1753

Section 2 is devoted to an illustration of the previous results through the analysis

of two examples of Bernoulli convolutions. We first study (2.1) the so-called(2, 3)-Bernoulli convolution: this measure is proved to be weak Gibbs (Theorem 2.4) and

we show that its Lq -spectrum displays a phase transition point (Theorem 2.5). The case of

the (2, 3)-Bernoulli convolution is closely related to our main application, concerned with

the multifractal analysis of the Erd os measure. The Bernoulli convolution (1 < < 2)

defined in 2.2 is a non-atomic probability measure supported by the unit interval which is

either continuous or purely singular (see [21]). Erdos proved in [10] that is purely

singular when is a Pisot number (i.e. an algebraic integer whose conjugates have

modulus less than 1). When = (1 +

5)/2 the measure := is called [42] theErdos measure. The multifractal analysis of has been partially studied in [13, 24, 27];

we prove (Theorem 2.9) that is a weak Gibbs measure (but not Gibbs) with respect

to a suitable 3-fold covering transformation (the potential of is defined by means ofcontinued fractions), so that the full multifractal formalism is completely established

(our contribution is concerned with the decreasing part of the singularity spectrum of ).

In [13], the first author completes a result in [24] by giving an explicit formula for the

Lq -spectrum of the Erdos measure, proving in addition that there exists a negative qc such

that: (i) (q) = q log2/log for any q qc; (ii) is infinitely differentiable at anyq > qc; and (iii) is not differentiable at qc. The weak Gibbs property of makes

possible (Theorem C) the use of the thermodynamic formalism and one may interpret qc

as a first-order phase transition. The variational principle allows an alternative approach

to (i) (Theorem 2.10), which is actually enough to ensure that is not real-analytic at

qc < 0; the fact that is not differentiable at qc is more difficult to establish: Appendix A

is devoted to a self-contained proof of this result based on the original argument in [13].

We point out that the multifractal formalism is valid for n , when n is the Pisot number

such that nn = n1n + + n + 1 (n 3) [13, 32], the corresponding Lq -spectrumbeing differentiable on the whole real line [13]. Even if some partial results can be

achieved (see e.g. [14]), the general case of a Pisot number seems to remain a difficult

problem.

1. Multifractal formalism of weak Gibbs measures

1.1. General framework. Let T be an expanding continuous transformation ofS1 such

that T1{x} is of cardinality d > 1 for any x S1. Assuming that T (0) = 0, thereexist d points xd = 0 = x0, x1, . . . , xd1 such that T is a one-to-one, onto mappingfrom [i] := [xi , xi+1[ to S1. The bilateral restrictions Ti :]xi , xi+1[ S1\{0} arediffeomorphisms with Holder-continuous derivative (the transformation T may not be

differentiable at the points xi ): we say that T is a regular d-fold covering transformation

(d-f.c.t.) ofS1. Let d be the one-sided direct product of an infinite number of copies of

the alphabet Ad := {0, . . . ,d 1}, i.e. d :=

0 Ad; we assume that d is endowed

with the product topology and : d d denotes the (one-sided) shift transformation.Each x S1 is associated to a unique point (x) := (k ) d such that Tk(x) [k],for any integer k 0, and : S1 d is a one-to-one map such that the following



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diagram commutes

S1T //

S1

d

// d

(we usually call the coding map of T). For any x S1 such that = (x), wedenote by In(x) = [0 n1] the n-step basic interval about x, that is, the set ofpoints y S1 such that Tk(y) [k], for k = 0, . . . , n 1. Let [[w]] be the set ofthe d such that 0 n1 = w, which is usually called the n-step cylinder set aboutw (clearly, 1([[w]]) = [w]). IfM0 := {S1} and ifMn (n > 0) is the collection ofthe n-step basic intervals then M := {M}n=0 is by definition the Markovian net of T.

The Holder-continuous function 0 : d R

such that 0() = log |T(x)| when = (x) (and extended to d by continuity) is called the volume-derivative potential ofT with respect to M. By a classical application of the Mean Value Theorem, there exists

a constant K > 1 such that, for any (S1) and any integer n > 0,1

K |[1 n]|

exp(Sn0()) K (4)

(|J| stands for the length of any interval J S1 and Sn0() :=n1

k=0 0(k )).

Definition 1.1. The measure defined on S1 is said to be a weak Gibbs measure of the

potential : d R, if there exists a sub-exponential sequence of real numbersK(n) > 1 (i.e. limn(1/n) log K(n) = 0) such that, for any n > 0, and any (S1),

1

K(n) [0 n1]

exp(Sn()) K(n); (5)

without loss of generality, we assume that K(n) increases with n.

If K(n) is constant, one recovers the classical notion of Gibbs measures [2];

accordingly, by (4), the Lebesgue measure is a Gibbs measure of the volume-derivative

potential 0.

Suppose that the probability measure is fully supported by S1; for (S1), we set1() := log [0] and for any n > 1,

n() := log[0 n1]

[1

n

1

]. (6)

Using the density of (S1) in d, one extends n to a continuousmap defined on the whole

ofd and called the n-step potential of. The following lemma provides a useful way to

prove that is weak Gibbs (the proof is left to the reader).

LEMMA 1.2. Letn be the n-step potential of a fully supported probability measure ; if

n converges uniformly to a potential then is a weak Gibbs measure of.

Even if a weak Gibbs measure need not be invariant under the dynamics ofT, the notion

is closely related to the theory of equilibrium states. The topological pressure of a potential



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:d

R (simply assumed to be continuous) is, by definition,

P() := limn

1

nlog

wAn

d

exp(Sn[[w]]), (7)

where Sn[[w]] := max{Sn(); [[w]]} (a sub-additive argument ensures that the limit

in (7) does exist). The variational principle of Walters [45] asserts that, for any -invariant

probability measure of metric entropy h(), one has h() + () P(), equalitybeing obtained when is an equilibrium state of ; the set of equilibrium states of is a

non-empty weak- compact convex set (in fact a Choquet simplex), whose extreme pointsare -ergodic measures. We shall use the basic properties of the topological pressure listed

in [45, Theorem 9.7].

It is worth noting that the weak Gibbs property is satisfied by a g-measure in the sense of

Keane [22] as well as the so-called conformal measures. More precisely, given a potential

: d R, the probability measure defined on S1 is said to be e -conformal, if forany Borel set A, with A [i] for some i Ad, one has

T (A) =

A

e(x) d(x). (8)

Under the condition that is fully supported by S1, it is easily seen that (8) implies the

uniform convergence of the n-step potentials of, ensuring by Lemma 1.2 that is weak

Gibbs. However, the converse is not true in general, even if there exists a partial reciprocal;

to see this, we notice that if in addition to being of full support and e -conformal,one also assumes that is T-invariant, then, according to the previous remark, is a

weak Gibbs measure of , but it is also necessary that is normalized in the sense that

=

e()

1 (the n-step potentials are trivially normalized and, using the uniform

convergence, one deduces that is also normalized). Now, if one assumes that is a

T-invariant weak Gibbs measure of the normalized potential , then it is easily seen that

is an equilibrium state of and by the variational principle in [26, Theoreme 1], one

deduces that is e -conformal. When the potential is normalized one usually writesg = e so that the T-invariant e -conformal measures are exactly the g-measures.

1.2. Statement of the multifractal theorems. The classical starting point of the

multifractal analysis is to make possible an application of the ShannonMcMillan

Theorem, by the introduction of the Markovian local dimension

DI M(x|M) := limn

log (In(x))

log|In(x)

|, (9)

provided that the limit exists; then the -level set corresponding to this local dimension is

by definition

E(|M) := {x S1; DI M(x|M) = }. (10)From now on, suppose that the probability measure defined on S1 is a weak Gibbs

measure of the negative potential : d R. For any (q,t) R R we consider thepartition function

Zn(q,t) :=

wAnd

[w]q|[w]|t . (11)



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Using the Gibbs property (4) of the Lebesgue measure and the weak Gibbs property (5)

of, it follows from the definition given in (7) that

P(q t 0) = limn

1

nlogZn(q,t) =: P (q,t). (12)

The convex property of P implies that P is a convex map on R R; for any q, q Rwith q > q, one has

q t0 (q q) sup() + (q t0),where sup() := sup{(); d}; hence

P (q t0) (q q) sup() + P(q t0),and thus

P(q,t)P(q

, t )q q sup(). (13)

Since is supposed continuous and negative, sup() < 0, and one deduces from (13)

that the convex map q P (q,t) decreases from + to . Using the assumptionthat sup(0) < 0, the same argument shows that, for any q R fixed, the convex mapt P (q,t) increases from to +. Therefore, there exists an increasing concavemap : R R defined by the implicit equation P(q,(q)) = 0. The underlyingthermodynamic formalism related to the convex map P leads to a characterization of the

points where is not differentiable (i.e. the first-order phase transition points): actually,

according to Lemma 3.2 proved in 3.1,

(q) = sup

()

(0

) and (q+) = inf ()

(0

) ,where the infimum and the supremum are respectively taken over the probability measures

, equilibrium states of the potential q(q)0. Lemma 3.2 also provides a descriptionof the behavior of about + and : more precisely we shall prove that

inf

()

(0)

= lim

q+(q)

q=: and sup

()

(0)

= lim

q (q)

q=: ,

where the infimum and the supremum are respectively taken over the probability measures

which are -invariant on d.

It is now possible to state the multifractal formalism of a weak Gibbs measure with

respect to the Markovian local dimension.

THEOREM A. LetM be the Markovian net of a d-f.c.t. T and be a weak Gibbs measure

of a negative potential : d R. The set of all points with E( |M) = is theinterval [, ] anddimH E( |M) = (), for any < < .

When T is differentiable on the whole torus and is a Gibbs measure associated to a

potential which is continuous with respect to the natural topology on S1, it is well known

[36] that E() = E(|M). We emphasize that, in our framework, the transformationT (respectively the potential of the weak Gibbs measure) may not be differentiable

(respectively continuous) for the natural topology on S1: in that case one may have

E() = E(|M). The following theorem is a crucial point of our multifractal analysis.



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THEOREM B. LetM be the Markovian net of a d-f.c.t. T and be a weak Gibbs measure.

Then, dimH E(|M) = dimH E(), for any R.Definition 1.3. The Lq -spectrum of is the concave map : R R {} such that

(q) := lim infr0

log inf

i (Bi )q; {Bi}i

log r

,

where {Bi}i runs over the family of covers ofS1 by closed balls of radius r .THEOREM C. Let T be a d-f.c.t. and be a weak Gibbs measure of a negative potential

: d R. Then, (q) = (q), for any q R.Finally, we can state the strong version of Theorem A.

THEOREM A . LetT be a d-f.c.t. and be a weak Gibbs measure of a negative potential : d R. The domain of the finite values of dimH E() is DO M() = [, ] anddimH E() = (), for any < < .

2. Bernoulli convolutions

For practical reasons, we shall need basic notions about the set of words on an alphabet.

GivenA := {0, . . . , s 1} (s 2) a finite alphabet, each element in An (n 1) is denotedby a string of n letters/digits in A that we call a word; by convention A0 is reduced to

the empty word . By definition, A is the set of words on A, that is, A := n=0 An.We denote by wm the concatenation of the two words w and m so that A, endowed withthe concatenation, is a monoid with unit element . Whenever x0, . . . , xs1 are s elementsof a monoid (X, ) with identity element e, we denote x

:=e and xw

:=x0

xn

1

,

for any word w = 0 n1 A.

2.1. The (2,3)-Bernoulli convolution. Let b and d be two integers with 2 b < d (b isfor basis and d is for digit). The (uniform) (b,d)-Bernoulli convolution is defined as the

probability distribution of the random variable X : d R such that

X() = 1b

b 1d 1

n=0

n

bn,

for any = (i )i=0 d, when d is endowed with the equidistributed Bernoulliprobability. The measure is non-atomic, is supported by the whole unit interval

I = [0, 1], and is either absolutely continuous or purely singular (see [21, Theorem 35]).In this section we focus our attention on the (2, 3)-Bernoulli convolution , when 3 isendowed with the equidistributed Bernoulli measure 3 (by [8, Propositions 5.2 and 5.3],

is known to be purely singular). The measure turns out to be self-similar; to see this,

notice that for M I one has X() M if and only if 0/4 + X()/2 M, that is,S0 X () M, where S0 (x) := x/2 + 0/4; since 3 is a Bernoulli measure, for = 0, 1 or 2 one obtains

3([[]] {S X () M}) = 3[[]]3{S X () M}= 1

33{S X() M} = 13 (S1 (M)).



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Finally, one deduces that satisfies the following self-similarity equation:

= 13

S10 + 13 S11 + 13 S12 . (14)Each dyadic sub-interval of I is coded by a word w {0, 2} and in what follows wedenote [w] := Sw (I ). From the self-similarity property of the measure given in (14) andthe fact that S11 S00(M) = {0} and S12 S00(M) = , for any M I, one obtains

[00w] = 13

(S10 [00w]) + 13 (S11 [00w]) + 13 (S12 [00w])= 1

3 (S10 S0[0w]) + 13 (S11 S00[w]) + 13 (S12 S00[w]) = 13 ([0w]).

Likewise, using the identity S12 = S20 and the fact that S10 S20(M) {1}, for any M I,[20w] = 13 (S10 S20[w]) + 13 (S11 S20[w]) + 13 (S12 S20[w])

=13 (S

11 S12[w]) +

13 [0w]

= 13

([2w]) + 13

([0w]),and the following matricial identity holds:

[00w][20w]

= 13 P0

[0w][2w]

, where P0 :=

1 0

1 1

. (15)

Since S10 = S02, one gets in the same way that[02w][22w]

= 13 P2

[0w][2w]

, where P2 :=

1 1

0 1

. (16)

A simple induction using (15), (16) and the fact that [0] = [2] = 1/2 yields, for any0 n1 {0, 2}

n

,

[0 n1] =1

2

1

3n1tV0 P0n1 V , (17)

where

V0 :=

1

0

, V2 :=

0

1

, V :=

1

1

.

In order to fit our framework, the measure is identified to a measure on the torus S1.

Let T2 : S1 S1 be the multiplication by 2 (mod 1); it is a 2-f.c.t. coded by the full shift : where := 0 {0, 2} and the volume-derivative potential is 0 : Rsuch that 0 log 2. A dyadic interval [0 n1] becomes an n-step basic interval,i.e. [0] = [0, 1/2[, [2] = [1/2, 1[ and x [0 n1] if and only if Tk2 (x) [k]whenever 0 k n 1.

Denote by 0 := (i = 0)i=0; then, for any potential : R,

exp(Sn(20)) =exp((20))

exp((0)){exp((0))}n.

A direct computation using (17) gives [20n1] = n/(2 3n1), for any n > 0; if is aGibbs measure of, then there exists a constant K > 1 such that, for any n 1,

1

K 1

n{3 exp((0))}n K;



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this is impossible and one concludes that is not a Gibbs measure. Our aim is to

prove that satisfies the weak Gibbs property with respect to some potential to beidentified. We consider the probability measure defined on S1 by setting, for any word0 n1 {0, 2}n,

[0 n1] =1

2

1

3ntVP0n1 V . (18)

It is clear that is T2-invariant and the next proposition shows how it is related to .

PROPOSITION 2.1. For any , and any integern 1,3

n + 2 [0 n1][0 n1]

3.

Proof. Let w {0, 2}n; it is easily seen that [w]/[w] 3 and thus it remains to provethat [w]/[w] 3/(n + 2). Assume that w = 0a1 2a2 ak , with {0, 2}, k 2 anda1, . . . , ak > 0 (the cases a1 = 0 or n are similar); if we denote

Pa21

2 Pak V =:

p

q

,

then

[w][w] =

3(1 1)

p

q

(1 1)Pa1

0 P2

p

q

= 3(p + q)(1 + a1)(p + q) + q

3a1 + 2

.

Since n = a1 + + ak, it follows that a1 n and thus [w]/[w] 3/(n + 2). 2In order to define the potential associated to in Theorem 2.2 below (as well as for the

Erdos measure in 2.2), we introduce some notations and ideas.

Given a sequence a0, a1, . . . of integers with a0 0 and ai > 0 for i > 0, we denote

[a0; a1, . . . , ak] := a0 +1

a1 +1

. . . + 1ak

=: pkqk

,

where the irreducible fraction pk /qk is the kth convergent of the continued fraction[a0; a1, . . . ] = limk[a0; a1, . . . , ak]. The integers pk and qk satisfy a well-known linearrecurrence, say

pk

qk

=

akpk1 + pk2akqk1 + qk2

= Qa0 Qak

1

0

where Qai :=

ai 1

1 0

; (19)

we refer to [23] for a general presentation of continued fraction theory.

For {0, 2} and a any positive integer, we denote |a = a; given any sequencea1, a2, . . . , ak (k 2) of positive integers, we define |a1, a2, . . . , ak by means of the



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induction formula

|a1, a2, . . . , ak

=a1

2

|a2, . . . , ak

. Let 0

:=(i

=0)

i

=0,

2 := (i = 2)i=0 and : 3 N {} such that

() =

min{k 0; k [[1]] {0, 2}} ifk 0, k [[1]] {0, 2}; ifk 0, k / [[1]] {0, 2};

hence the map : 3 N {} is the hitting time of [[1]] {0, 2}. Moreover, if() = 0 then we set n() = 0, and if 0 < () < , there exists a unique finitesequence of n() positive integers a1 , . . . , a

n()

with a1 + + an() = () and suchthat, 0 ()1 = 0|a1 , . . . , an(); when () = we set n() := andthere exists a unique (infinite) sequence of integers a1 , a

2 , . . . with a

i > 0 and such that

0 n1 = 0|a1 , . . . , ak , , where 0 < ak+1 and a1 + + ak + = n.

THEOREM 2.2. The T2-invariant probability measure defined on S1

by (18) is ag-measure of the normalized potential : R such that

() =

log(1/3), ifn() = () = 0,log([1; a1 , . . . , an()]/3), if0 < n(), () < ,log([1; a1 , . . . ]/3), ifn() = () = .

The proof of Theorem 2.2 relies on the following lemma, which, according to (19),

makes the link between the matrix product formula in (18) defining and the continuedfractions involved in the definition of the potential associated to in Theorem 2.2.

LEMMA 2.3. Forw = |a1, . . . , ak, one has tVPwV = tVQa1 Qak V.

Proof. Notice that Q0Q0 is the identity matrix and that Pa2 Q0 = Q0Pa0 = Qa , for eachinteger a 0; given any finite sequence of integers a1, . . . , ak with a2 ak1 > 0, onehas

tVPa1

2 Pa2

0 Pak2 V = tV (Pa12 Q0)(Q0Pa20 ) (Pak2 Q0)Q0V= tVQa1 Qa2 Qak V . 2

Proof of Theorem 2.2. Assume that () = (the case () < being similar).For n 1 one writes 0 n1 = 0|a1 , . . . , ak , , with n = a1 + + ak + and0 < a

k+1; one considers x := [1; a1 , . . . ] and for any k 1,

pk

qk= [1; a1 , . . . , ak ] and pkq k

:= [1; a1 , . . . , ak , + 1].

By the definition of the n-step potential n of, it follows from Lemma 2.3 that

3 exp(n()) =tVQa1

Qa2 Qak Q V

tVQa11Qa

2 Qak QV

=(1 0)Q1Qa1

Qak Q+1 t(1 0)(0 1)Q1Qa

1 Qak Q+1 t(1 0)

= (1 0)t(pk q

k )

(0 1) t(pk qk )

= pk

q k.



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Using classical inequalities of the theory of continued fractions,x pkq k

x pkqk+pkqk

pkq k

1qk qk+1 +1

qk qk

1qk+1

+ 1q k

.

Since n = a1 + + ak + , one gets qk+1 q k > n, which gives the following upperbound:

|exp(()) exp(n())| 2

3n.

The desired result is obtained by an application of Lemma 1.2. 2

From Proposition 2.1 and Theorem 2.2, one deduces the following.

THEOREM 2.4. The (2, 3)-Bernoulli convolution is a weak Gibbs measure of.

Since is a weak Gibbs measure of the potential , it satisfies the multifractal

formalism as stated in Theorem A; moreover, by Theorem C, its Lq

-spectrum coincideswith the concave function , a solution of the implicit equation P (q,(q)) = 0.The following theorem proves that is not real-analytic at a critical point qc < 0, meaning

that qc is a phase transition point.

THEOREM 2.5. The Lq -spectrum of the (2, 3)-Bernoulli convolution is a concave

function for which there exists qc < 0 such that(q) = q log3/log2 if and only ifq qc.The proof of Theorem 2.5 depends on the following lemma.

LEMMA 2.6. limq

k=1

a1,...,ak >0( tVQa1 Qak V )q = 0.

Proof. We first prove that ifa1, . . . , ak are positive integers then

tVQa1 Qak V a1 + 22

ak + 22

, (20)

where = (1 +

5)/2. It is readily checked that tVQa V (a + 2)/2 when a > 0.Given k positive integers a1, . . . , ak , one has, for any integer a > 0,

tVQa1 Qak QaV tVQa1 Qak1 Qak1

a + 1a + 1

.

If (20) is valid for rank 1 up to rankk 1, then (even when ak 1 = 0),tVQa1 Qak Qa V

a1 + 2

2

ak1 + 2 2

ak 1 + 2

2

(a + 1)

a1 + 2

2

ak1 + 2

2ak + 2

2a + 2

2 ,

and (20) follows by induction. Using (20), one gets, for any q < 0,

k=1

a1,...,ak >0

( tVQa1 Qak V )q

k=1

a1,...,ak >0

a1 + 2

2

ak + 2 2

q

k=1

n=1

n + 2

2

qk

and the desired result holds. 2



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Proof of Theorem 2.5. The Bernoulli convolution being a weak Gibbs measure of, one

deduces from Theorem C that (q) = (q): this means that, for any q R,(q) = max{t;P(q,t) 0} = min{t;P (q,t) 0}. (21)

Let 0 := log3/log2; for 0 = (i = 0)i=0 one has 0(0) = log 2 and (0) = log3,so that q(0) 0q0(0) = 0, for any q R: it follows from the variational principlethat

0 = h(0) +

{q() 0q0( )} d0( ) P(q 0q0) =: P (q,0q),

which by (21) implies that (q) 0q . We now prove that (q) 0q when q issufficiently close to . Since is also a weak Gibbs measure of , one can write

P(q,t) = limn

1

nlogZn(q,t), where Zn(q,t) :=

w{0,2}n

[w]q /|[w]|t.

By Lemma 2.3, one has, for any q < 0,

Zn(q,0q) :=

w{0,2}n([w]/|[w]|0 )q

= 12q

w{0,2}n

( tVPwV )q = 2

2q

nk=1

a1++ak=n

( tVQa1 Qak V )q

so that

Zn

(q,0

q)

2

2q

k=1

a1,...,ak>0

( tVQa1

Qak

V )q .

Hence, by Lemma 2.6, there exists q0 < 0 such that Zn(q, q ) 1/2q for any q < q0 andeach n 1. Therefore, P (q,0q) 0 and (21) gives (q) 0q when q q0; since is a concave map with (0) = 1, there exists qc < 0 such that (q) = 0q if and only ifq qc. 2

2.2. The Erd os measure. Suppose that 1 < < 2 and let := 1/(1). The Bernoulliconvolution can be defined as the probability distribution of the random variable

Y : 2 R such that Y () = (1/)

k=0 k /k+1, where 2 is endowed with

the equidistributed Bernoulli measure. As in the case of a (b,d)-Bernoulli convolution,

satisfies a self-similarity equation, say

= 12 S10 + 12 S11 , (22)

with S(x) = (x + /)/ and {0, 1}. From now on, we assume that = (1 +

5)/2,

that is = is the Erdos measure. The algebraic equation 2 = + 1 satisfied by implies that 1/ = 1 =: , so that S0(x) = x and S1(x) = x + 2. It is easily seenthat the intervals S00[0, 1[, S100[0, 1[ = S011[0, 1[ and S11[0, 1[ form a partition of[0, 1[;this means that R0 := S00, R1 := S100 = S011 and R2 := S11 define a non-overlappingsystem of affine contractions on [0, 1[ and for any word w on the alphabet {0, 1, 2}, we



13/34


denote

[w

] =Rw(I ). By symmetry

[0

] =

[2

]and according to the self-similarity

property (22) satisfied by , one gets [1] = ([0] + [2])/2, so that[0] = [1] = [2] = 1

3. (23)

Likewise, for any word w on the alphabet {0, 1, 2},

[1w] = 12

(S10 [1w]) + 12 (S11 [1w])= 1

2(S10 S011[w]) + 12 (S11 S100[w])

= 12

(R2[w]) + 12 (R0[w]) = 12 [2w] + 12 [0w],

which we can write in the following matricial way:

[1w][1w] = P1[0w][2w] where P1 := 1/2 1/21/2 1/2 . (24)As initially noticed by Strichartz et al. [43], the identity S100 = S011 plays a crucial role inthe overlapping situation involved in the self-similarity of. It follows from the same kind

of elementary computation which yields (24) that[01w][21w]

= 1

4P1

[0w][2w]

; (25)

[00w][20w]

= 1

4P0

[0w][2w]

where P0 :=

1 0

1 1

; (26)

[02w][22

w] =

1

4

P2 [0w][2

w]

where P2

:= 1 1

0 1 . (27)

By induction using (23), (24), (25), (26) and (27), one gets for any 0 n1 {0, 1, 2}n,

[0 n1] =1

3

1

4n1tV0 P0 Pn1 V , (28)

where

V0 :=

1

0

, V2 :=

0

1

, V1 =

1/2

1/2

.

Similarly to the (2, 3)-Bernoulli convolution, the Erdos measure is considered as a

probability measure on S1; moreover, the transformations R0, R1 and R2 are identified to

the local inverses of a 3-f.c.t. ofS1 denoted T which is coded by the full shift : 3 3and associated to the volume-derivative potential 0

:3

R, such that

0() = log 21[[0]]() + log 31[[1]]() + log 21[[2]]().

We now consider the intervals [w] (w {0, 1, 2}) as subsets of S1 in such a way that[0] := [0, 2[, [1] := [2, [, [2] := [, 1[ (defining the one-step basic intervals ofT) andfor each word 0 n1 {0, 1, 2}n (n 1)

x [0 n1] Tk(x) [k] k = 0, . . . , n 1,

(defining the n-step basic intervals ofT).



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By (28) one gets that

[10n

] =(2n

+4)/(3

4n+1) and thus is not a Gibbs measurewith

respect to T. Following the same idea as in the case of the (2, 3)-Bernoulli convolution,we introduce the T-invariant probability measure , such that

[0 n1] =1

2

1

4ntVP0 Pn1 V . (29)

The same argument leading to Proposition 2.1 gives the following.

PROPOSITION 2.7. For any 3, and any integern 1,2

n + 2 [0 n1][0 n1]

4.

In order to establish the following theorem, we use formula (29) defining togetherwith Lemma 2.3 and we apply the argument of the uniform convergence of the n-step

potentials (Lemma 1.2) in a similar way leading to Theorem 2.2.

THEOREM 2.8. The T-invariant probability measure defined on S1 by (29) is ag-measure of the normalized potential : 3 R such that

() =

log(1/4), ifn() = () = 0,log([1; a1 , . . . , an() + 1]/4), if0 < n(), () < ,log([1; a1 , . . . ]/4), ifn() = () = .

As a corollary of Proposition 2.7 and Theorem 2.8, one has the following.

THEOREM 2.9. The Erd os measure is a weak Gibbs measure of.

The case of the Erdos measure is similar to the one of the (2, 3)-Bernoulli

convolution studied in 1.1; since is a weak Gibbs measure of the potential , itsatisfies the multifractal formalism as stated in Theorem A and, by Theorem C, itsLq -spectrum coincides with the concave function , a solution of the implicit equation

P(q,(q)) = 0. Theorem 2.10 below is the analog of Theorem 2.5: it proves that isnot real-analytic at a critical point qc < 0, meaning that qc is a phase transition point.

THEOREM 2.10. The Lq -spectrum of the Erd os measure is a concave function for which

there exists a real numberqc < 0 such that(q) = q log2/log if and only ifq qc.Proof. The Erdos measure being a weak Gibbs measure of , one deduces from

Theorem C that (q) = (q): this means that, for any q R,(q) = max{t;P(q,t) 0} = min{t;P (q,t) 0}. (30)

For 0 := log2/log one can check that q (0)0q0(0) = 0, for any q R: it followsfrom the variational principle that

0 = h(0) +

{q() 0q0( )} d0( ) P(q 0q0) =: P(q,0q),

which implies that (q) 0q . We now prove that (q) 0q when q is sufficientlyclose to . Using the fact that is also a weak Gibbs measure of , it is easily seen that

P(q,t) = limn

1

nlog Zn(q,t) where Zn(q,t) :=

w{0,1,2}n

[w1]q|[w1]|t . (31)



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Notice that any word w

{0, 1, 2

}n is associated to a unique sequence of (possibly empty)

words m1, . . . , mk (1 k n) on the alphabet {0, 2} such that w1 = m11 mk1; since[w1] = [m11] [mk1] and |[w1]| = |[m11] | | [mk1]|, one has

Zn(q,t)

k=1

m1,...,mkmi {0,2}

[m11]q|[m11]|t

[mk1]q|[mk1]|t

k=1

m{0,2}

[m1]q |[m1]|t

k

(recall that {0, 2} := {}n=1{0, 2}n). However, according to Lemma 2.3, one gets

m{0,2}

[m1]q|[m1]|0q = 2

q + 2

k=1

a1ak >0

( tVQa1 Qak V )q ,

which implies that Zn(q,0q) is bounded for q < q0, with a small enough q0 given byLemma 2.6: by (31) one has P(q,0q) 0 and thus (q) 0q when q q0; since is concave with (0) = 1, there exists qc < 0 such that (q) = 0q if and only ifq qc. 2

Remark 2.11. (1) Starting from Definition 1.3 of , it is proved in [13] that is not

differentiable at qc; according to our terminology and the fact that = , this meansthat qc is a critical value of a first-order phase transition: we include in Appendix A a

proof of this result, based on the approach developed in [13].

(2) The phase transitions occurring in the multifractal formalism of the (2, 3)-Bernoulli

convolution and the Erdos measure are to be related to the problem of phase transition on

one-dimensional lattice systems and one-sided full-shift respectively studied in [9, 15] and

[18] (see also [19] and [30, 31] for the relationship with the multifractal formalism).

(3) We point out the similarity between the potentials associated with the (2, 3)-

Bernoulli convolution (Theorem 2.2), the Erdos measure (Theorem 2.8) and the potentials

considered in [41].

(4) The Lq -spectra (for q > 0) and the sets of possible local dimensions for a special

class of self-similar measures with overlaps were studied in [12, 20]; the Hausdorff

dimension of the corresponding self-similar sets was determined in [38].

3. Proof of the multifractal theorems

3.1. Proof of Theorem A: lower bound. For any probability measure , -invariant

on d, we denote by G

() the set of -generic points of d, meaning that

G

()

if and only if Snf ()/n tends to (f), for any real-valued continuous function f defined

on d. Since the full-shift d satisfies the specification property, the set G() is never

empty (especially when is not ergodic).

PROPOSITION 3.1. LetT be a d-f.c.t. ofS1 and : S1 d the associated coding map;for any -invariant measure (not necessarily ergodic) on d, one has

dimH 1(G()) =

h()

(0).



16/34


Hint. For any Borel set M

d, we denote by (M) the Lebesgue measure of

1(M);the probability is non-atomic and fully supported by d: by setting for any , d,

d(,) =

1, if0 = 0,inf{[[0 n1]]; [[0 n1]]}, if0 = 0,

one defines a metric d compatible with the product topology on d. Let dim be the

-Billingsley dimension [1], that is, the Hausdorff dimension on the metric space (d, d).

Given a probability measure -invariant on d, we claim that

dim G() = h()

(0). (32)

When is ergodic, this formula can be deduced by a classical argument using theShannonMcMillan Theorem and the fact that (G()) = 1. However, (32) is stillvalid when is -invariant without being ergodic: this follows from [5, Theorems 7.1 and

7.2] and [29, Theoreme 2], where the fact that is a Gibbs measure is needed.

Actually, it is easily seen from (4) that is a Gibbs measure of0 in the sense that there

exists a constant K > 1 such that, for any d and any integer n 1,1

K [[0 n1]]

exp(Sn0()) K;

then we can apply [37, Theoreme 1.2.2], ensuring that dimH 1(M) = dim M, for any

M d: according to (32), one concludes that Proposition 3.1 holds.

We now prove a lemma that gives the characterization of the points where is not

differentiable (i.e. the first-order phase transition points); it is essentially a corollary of

the tangency property of the topological pressure [45, Theorem 9.15] saying that is an

equilibrium state of the potential : d R if and only ifP ( + ) P() (),for any real-valued continuous function defined on d.

LEMMA 3.2. Given a negative potential : d R and the concave map such thatP(q,(q)) = 0, the following two propositions hold:(i) ifIq is the simplex of the equilibrium states of q (q)0 then

(q)

=sup

Iq ()

(0) and (q+) = infIq

()

(0) ;

(ii) ifI(d) is the simplex of the -invariant probability measures on d then

infI(d)

()

(0)

= lim

q+ (q)

q=:

and

supI(d)

()

(0)

= lim

q(q)

q=: .



17/34


Proof. (i) Let q > 0; from the definition of the concave map

:R

R one has

P(q (q)0) = 0; similarly, for any h > 00 = P((q + h) (q + h)0) = P(q (q)0 + h( ((q+) + h)0)),

where we write (q + h) = (q) + h (q+) + hh with h tending to 0 when h tendsto 0. In conclusion, one has the equation

P(q (q)0 + h( ( (q+) + h)0)) P(q (q)0) = 0;which, by the tangency property of the pressure, ensures that

() ( (q+) + h)(0) 0,for every Iq , that is, (q+) ()/(0) h. When h tends to 0, one gets

(q+) ()/(0) and thus(q

+) inf

()

(0); Iq

.

By the same argument one gets

sup

()

(0); Iq

(q).

It is clear that if is differentiable at q then (q) = ()/(0), for every Iq .

Since is not differentiable on an (at most) countable subset ofR, there exists a sequence

of real numbers qn > q which tend to q and such that is differentiable at qn for any n.

Let n Iqn and assume (by compactness) that n tends to a -invariant probabilitymeasure in the weak- sense; since the potentials qn (qn)0 converge uniformlyto q (q)0, it follows from the variational principle and the upper semi-continuityof the entropy that Iq . Using the fact that is concave,

(q+) = lim

n(qn) = limn

n()

n(0)= ()

(0),

and then (q+) = inf{()/(0); Iq}. The same argument applies to (q).

(ii) We prove the assertion for (the same argument applies to ). By the concavity of

it is clear that limq+ (q+); this fact together with part (i) yields

inf

()

(0); I(d)

.

Moreover, if I(d) then, by the variational principle, h() + (q (q)0) P(q,(q)) = 0, for any q > 0. Since 0 is negative and uniformly bounded awayfrom 0,

(q)

q h()

q(0)+ ()

(0) ()

(0).

Therefore = limq+ (q)/q ()/(0), that is,

inf

()

(0); I(d)

. 2



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LEMMA 3.3. For any < < , letq

R such that(q+ )

(q ); then there

exists Iq such that()/(0) = and for such a measure one has h()

(0)= sup{q R; q (q)} =: ().

Proof. We consider the map : ()/(0) which is continuous on the convex setof probability measures on d, endowed with the weak- topology. Given < < ,the set (Iq ) is a non-empty closed interval ofR, for Iq is a non-empty closed convex

subset of the -invariant measure. By Lemma 3.2, [(q+ ),(q )] = (Iq ) and thusthere exists Iq (not necessarily ergodic) such that () = ()/(0) = .We now apply the variational principle: on the one hand, for every q R,

h() + (q (q)0) P(q (q)0) = 0;

on the other hand, since Iq ,h() + (q (q)0) = P (q (q)0) = 0.

Therefore, h()/(0) = q (q) and h()/(0) q (q), for anyq R, which means, by definition, that () = h()/(0). 2

We are now in a position to prove the lower bound () dimH E(|M), for < < . Given x S1 with (x) = , the Gibbs property of the Lebesgue measure(with respect to the volume-derivative potential 0) ensures the existence of C > 1 such

that, for any integer n,

Sn0() log C log |In(x)| Sn0() + log C.Likewise, by the weak Gibbs properties of , there exists a sub-exponential sequence of

real numbers K(n) > 1 such thatSn() log K(n) log (In(x)) Sn() + log K(n).

The potential being negative, () < < 0 uniformly on d and Sn()/n < for

any n; moreover, since (1/n) log K(n) tends to 0 when n goes to infinity, one deduces

that Sn() + log K(n) < 0, when n is sufficiently large. For the same reason that 0 isnegative, one also has Sn0() + log C < 0, when n is sufficiently large, and thus

Sn() + log K(n)Sn0() log C

log (In(x))log |In(x))|

Sn() log K(n)Sn0() + log C

. (33)

Now assume that = (x) G(), where Iq (given by Lemma 3.3), satisfies()/(0) = ; by the definition ofG(), one has limn Sn()/n = () < 0 andlimn Sn0()/n

=(0) < 0, so that

limn

log (In(x))

log |In(x)|= ()

(0)= ;

the point x being arbitrarily taken in 1(G()), one deduces that

1(G()) E(|M).Using successively Lemma 3.2 and Proposition 3.1, one concludes

() = h()

(0)= dimH 1(G()) dimH E(|M).

The required lower bound is proved.



19/34


3.2. Proof of Theorem A: upper bound. The following proof is essentially the argument

given by Brown et al. [4] that we adapt to our framework. Notice that the hypothesisthat is a weak Gibbs measure of the potential is implicitly needed to ensure that the

concave map is well defined. We use the notions of-packing, box dimension (denoted

dimB ) and packing dimension (denoted dimP), presented in Appendix B. We shall prove,

in Theorem 3.4 below, a stronger result than the upper-bound dimH E(|M) ()involved in Theorem A; actually in place of the level sets E(|M), we shall deal withwhat we call the fat level sets, defined by setting, for any R,

F+(|M) :=

x S1 : lim supn

log (In(x))

log |In(x)|

and

F(|M) := x S1 : lim infn log (In(x))log |In(x)| .THEOREM 3.4. LetM be the Markov net of a regulard-f.c.t. T and be a weak Gibbs

measure of a negative potential : d R. Then, the following propositions hold:(i) if < (0) then dimP F+( |M) ();(ii) if (0

+) < then dimP F( |M) ().Proof. We prove part (i) while part (ii) can be handled in a similar way. See Figure 1

for the graph of (q). To begin with, notice that, for (0

+) (0), one has () = (0) = 1, so that the upper bound dimP F+( |M) () is trivial in thatcase. We now consider that < < (0

+); given such that < < (0+), we

define

Fn+() := {x S1; |In(x)| (In(x))},for any n 0, so that

F+(|M)

m=0

n=m

Fn+(). (34)

Let m 0 and > 0 be arbitrarily chosen; we consider J = n=m Jn, where Jn Andand such that {[w]; w J} is an -packing ofn=m Fn+(). By definition of an -packing,it is clear that, for any word w J, there exists x n=m Fn+() such that [w] = In(x)for some n m, so that |[w]| [w]. Accordingly, for any 0 and q > 0, one canwrite

wJ |[w]|

=

n=m

wJn|[w]|

q

/|[w]|q

n=m

wJn

[w]q /|[w]|q

n=1Zn(q,q ),

where Zn(, ) is as in (11). Since < (0+), there exists q0 > 0 with q0 (q0)= (). When > () one has q0 < (q0), which, by definition of , impliesthat

P (q,q0 ) = P (q0 (q0 )0) < 0.



20/34


t=

(q)

q0 () = (q0)

()

1

q0

1

q

(q0, q0 )q0

FIGURE 1. We represent here the graph of the concave map q (q) (with a discontinuity of the derivative atq = 0) and the gray area is the open domain of the ( q,t) RR for which P (q,t) < 0; when < < (0+),the straight line with equation t = q

() is tangent to the graph of the map at the point with abscissa

q0 > 0, and P (q0, q0 ) < 0 when > ().

Hence, there exists < 0 and a rank n0 0 such that Zn(q0, q0 ) exp(n)for any n n0; it follows that

wJ [w] < , for any > (). This proves

that dimB

n=m Fn+() (), for any m 0, and from (34) one concludes that

dimP F+(|M) (). Since is an arbitrary real number such that < < (0+),the continuity of at yields the desired upper bound stated in (i); the proof of thetheorem is complete. 2

3.3. Proof of Theorem B: upper bound. There exist conditionson thed-f.c.t. T ensuring

that E() = E(|M): for instance, this occurs when T is differentiable at the point 0 := 0(mod 1). In our setting, the inclusion E() E(|M) does not hold in general; however,we shall establish the upper bound dimH E()

()

=

dimH E(|M) (see part (iii)

of Proposition 3.5 below), by means of the upper bounds in Theorem 3.4 involving the fat

level sets F(|M) and F+(|M).PROPOSITION 3.5. Let M be the Markov net of a regular d-f.c.t. T; if is a Borel

probability measure then the following two properties hold:

(i) E() F(|M);(ii) dimH E() dimH F+( |M);if in addition is a weak Gibbs measure, then one has:

(iii) dimH E() ().



21/34


In order to prove Proposition 3.5 we use ideas taken from the approach of Hofbauer

in [19]. Let dist(, ) be the usual distance on S1; for any x S1 and 0 < d < 1/2, wedefine

Nd(x) := {n N; dist(Tn(x), 0) > d} and Md := {x S1; #Nd(x) < }(#A stands for the cardinality of the countable set A.)

LEMMA 3.6. [19, Lemma 13] Suppose thatT is a regulard-f.c.t. of the torus S1; then

limd0

dimH Md = 0.

We define the n-step variation of the volume-derivative potential 0, by setting

V0(0) := max{0() 0( ); , d},Vn(0) := max{0( ) 0(); 0 n1 = 0 n1} for n 1, (35)and we consider in addition

n(0) := V1(0) + + Vn(0). (36)Notice that, since 0 is continuous on d, the variation Vn(0) tends to 0 when n goes

to infinity and, by a classical lemma on Cesaro averages, n(0)/n tends to 0 as well.

We denote by dn(x) the distance of x to the boundary of the basic interval In(x); the

following lemma shows that the ratio |In(x)|/dn(x) is a sub-exponential sequence, whenx S1\Md, for some 0 < d < 1/2.LEMMA 3.7. Given any x

S1

\Md, there exists a rankn0 (depending on x) such that,

for any n n0,0 < log

|In(x)|dn(x)

n(0) + log d.

Proof. By an application of the Mean Value Theorem, it is clear that

log |In(x)| supyIn(x)

{Sn0 (y)};

using the fact that x / Md, another application of the Mean Value Theorem yieldslog dn(x) inf

yIn(x){Sn0 (y)} + log d,

and one concludes using the definition of n(0).2

Proof of Proposition 3.5. (iii) When is a weak Gibbs measure, the upper bound

dimH E() () is a trivial consequence of Theorem 3.4 together with parts (i) and(ii), which we now establish.

(i) Fix x S1 and set n := |In(x)|, for any n 0. Then, for any rankn 0,log (Bn (x))

log n log (In(x))

log |In(x)|,

ensuring that E() F(|M).



22/34


(ii) Since T is regular and uniformly expanding,

0 < := infd

{exp(0())} supd

{exp(0())} =: < 1

and, by a classical application of the Mean Value Theorem, n |In(x)| n, for anyx S1 and any integer n 0. Let x S1\Md and denote by dn(x) the distance ofx to the boundary of the basic interval In(x); it follows from the definition of Md that

Bdn(x) (x) In(x) and thus

log (Bdn(x)(x))

log dn(x) log (In(x))

log dn(x)= log (In(x))

log |In(x)|

1 + log(dn(x)/|In(x)|)

log |In(x)|

1. (37)

Since one clearly has (1/n) log |In(x)| log > 0, one deduces from Lemma 3.7

thatlog(dn(x)/|In(x)|)

log |In(x)|= (1/n) log(|In(x)|/dn(x))(1/n) log |In(x)|

(n(0) + log d)/nlog

,

which according to (37) yields

log (Bdn(x) (x))

log dn(x) log (In(x))

log |In(x)|

1 (n(0) + log d)/n

log

1,

and thus E() (S1\Md) F+(|M), for n(0)/n tending to 0. Let > 0 bearbitrarily given; by Hofbauers lemma there exists d > 0 such that dimH Md , sothat

dimH E() = dimH E() (S1

\Md) + dimH E() Md dimH F+(|M) + .Part (ii) is established since > 0 is arbitrarily chosen. 2

3.4. Proof of Theorem B: lower bound. In order to prove the lower bound ()

dimH E() we use the underlying Markov structure to construct a slim level set S( |M),a subset of E() having the specified Hausdorff dimension, say ()

. Usually this isachieved by constructing a Frostman measure on the level set E(); we shall use this

approach with the additional difficulty that the measure to consider may not give a positive

measure to E() (this is related to the fact that may correspond to a first-order phase

transition point).

The proof of the desired lower bound is a consequence of Theorem 3.8 below. Let us

split any d into an infinite sequence of finite words, say 1, 2, . . . , so that(by concatenation) = 12 and with the additional condition that the length ofeach word n (n 1) is exactly n. Then we define the one-to-one map : d d bysetting

() := 11121 1n1n+11 =: = (i )i=0. (38)THEOREM 3.8. Let be a -invariant measure on d (d > 2); then the following hold:

(i) G() := (G()) G(); If T is a 2-f.c.t. one can consider T T.



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(ii) dim G()

=dim G();

(iii) for any x S1 such that(x) (d) and any probability measure on S1,DI M(x |M) = DI M(x) = .

We claim that, for each < < , the upper bound () dimH E() is a

consequence of Theorem 3.8. To see this, let q R be such that (q+ ) (q+ );using Lemma 3.3, we consider an equilibrium state of the potential q (q)0, say, for which ()/(0) = so that 1(G()) E(|M) and

dim G() = h()

(0)= ().

We define the slim level set S(|M) := 1(G()); by part (iii) of Theorem 3.8,one has DI M(x

|M)

=DI M(x) for any x

S(

|M), and since 1(G())

E(|M), one deduces from part (i) of Theorem 3.8 that S( |M) E(); by part (ii)of Theorem 3.8, one concludes that

() = dim G() = dim G() = dimH S(|M) dimH E(),

completing the proof of the desired lower bound.

Therefore it remains to establish Theorem 3.8. To begin with we shall prove the

following lemma.

LEMMA 3.9. LetT be a regulard-f.c.t. ofS1 (d > 2) and a Borel probability measure.

Suppose thatx is a point ofS1 such thatTmk (x) [1] (k 1), where m1, m2, . . . form astrictly increasing sequence of integers; if limk log |Imk (x)|/log |Imk+1 (x)| = 1, then

DI M(x |

M) =

DI M(x) = .

Proof. Let x S1 such that Tmk (x) [1] (k 1); for = (x), it is clear thatImk+1(x) = [0 mk11] Imk (x) = [0 mk1].

The Lebesgue measure being Gibbs with respect to the Markov net associated to T, there

exists a constant 0 < c < 1 such that, for any i = 0, 1, 2,|[0 mk1i]| c|[0 mk1]| =: rk ,

ensuring that Brk (x) Imk (x). Moreover, Imk+p(x) Brk (x) for some constant integerp > 0 and thus the following sequence of inclusions arises:

Imk+

p(x)

Brk

(x)

Imk

(x)

Brk

/c(x).

IfDI M(x|M) = then it follows that log (Brk (x))/log rk tends to , when k goes to ;notice that for rk+1 r rk

log rk

log rk+1log (Brk (x))

log rk log (Br (x))

log r log rk+1

log rk

log (Brk+1 (x))

log rk+1,

and from the assumption that limk log |Imk (x)|/log |Imk+1 (x)| = 1, one gets DI M(x) = .Conversely, if DI M(x) = , a similar argument proves that DI M(x|M) = . 2 Here, we use again the fact [37, Theoreme 1.2.2] that dimH

1(M) = dim M, for any M d.



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Proof of Theorem 3.8. For any integer p

1, define np

:= p1k

=0 k

=p(p

1)/2 and

np := np + p = np+1. For d given, recall that () = = (i )i=0 and noticethat, by construction,

np [[p]], np1 [[1]] and np [[p]].

(i) Assume that f is a real-valued continuous function defined on d. For any

G() and any integers n, p with p 2 and 0 n < p, one hasSnp+nf (

) Snp+nf()

=p1k=1

{f (nk1) + Sk f (nk ) Sk f (nk )}

+ {f (

np1)

+Snf (

np )

Snf (

np )

} Spf (

np+n),

=p1k=1

{Sk f (nk ) Skf (nk )} + {Snf (n

p ) Snf (np )}

+p

k=1f (n

k1) Sp f (np+n),

which gives, with Vk(f ) defined as in (35) and n(f ) as in (36),

1

np + n|Snp+nf () Snp+nf()|

2

p(p + 1)

pk=1

kj=1

Vj(f ) + pV0(f )

2(p(f ) + V0(f))p

+1

.

With k (f ) := k (f ) + V0(f ), a straightforward computation yields, for any n 1,1

n|Snf () Snf()|

2n (f )n + 1

, (39)

where n is the integral part of (

8n + 1 1)/2. The variation Vn(f ) tends to 0 when ngoes to infinity, as well as both n(f)/n and

n(f)/n; this completes the proof of (i).

(ii) Let n := n(n 1)/2 + r, where r is an integer such that 0 r < n and letn = n + n. Since the Lebesgue measure is a Gibbs measure of the volume-derivativepotential 0, one gets

log |[0 n1]| Sn 0() log Kfor the constant K > 1 given in (4). Thereafter, from (39) one deduces that

log |[0 n1]| Sn0() + Sn 0(n) 2(n + n)n (0)

n 1 log K

Sn0()

1 n0n sup(0)

2(n + n)n sup(0)

n (0)n 1

log Kn sup(0)

,

with sup(0) := sup{0( ); d} < 0, that islog |[0 n1]| Sn0()(1 An), (40)



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where

An := n0n sup(0)

+ 2(n + n)n sup(0)

n (0)n 1

+ log Kn sup(0)

is clearly a quantity that tends to 0 when n goes to infinity. We use again the Gibbs property

of the Lebesgue measure, say,

log |[0 n1]| Sn0() + log K,

which yields the following sequence of inequalities:

log |[0 n1]|Sn0()

1 + log KSn0()

1 + log Kn sup(0)

. (41)

Since 0 < 0, it is clear that Bn:=

log K/(n sup(0)) tends to 0 when n goes to infinity

and from (41) (with n large enough),

Sn0() log |[0 n1]|/(1 + Bn). (42)

It follows from (40) and (42) that there exists a sequence of real numbers b1, b2, . . . with

limk bk = 0 and such that

log |[0 n1]| log |[0 n1]|

1 + An1 + Bn

. (43)

In conclusion, for any > 0, there exists > 0 such that

d(,) d(,)1+ d(, ), (44)

and one deduces that dim G() = dim G(), for any -invariant measure .

(iii) Let x S1 such that (x) = (d). For mk := nk 1 one has Tmk (x) [1](i.e. mk [[1]]) and limk log |Imk+1 (x)|/log |Imk (x)| = 1. The result follows by anapplication of Lemma 3.9. 2

3.5. Proof of Theorem C. Let F0 := {S1} and suppose that Fn (n > 0) is a partition ofS1 by non-empty intervals. We say that F := n Fn is a net if any interval in Fn is theunion of intervals in Fn+1. For x S1 we denote by In(x) the interval in Fn such thatx In(x), and we suppose that limn |In(x)| = 0. Moreover,F is said to be regularif thereexists a constant c > 1 such that, for any J, J with Fn+1 J J Fn (n 0), one has

|J|/|J| c. Given 0 < r < 1 and x S1

, letn

r (x) be the rank such that |Inr (x)(x)| < rand |Inr (x)1(x)| r; furthermore, one defines an equivalence relation on S1 by settingx

ry if and only ify Inr (x) (x): the so-called r-Moran partition Fr is the partition of S1

generated by this equivalence relation. Notice that, under the condition ofF to be regular,

any interval J Fr has approximately length r, in the sense that

r/c |J| < r.

We refer to the book of Pesin [35] for a systematic presentation of the previous framework.

In order to prove Theorem C we first establish the following theorem.



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THEOREM 3.10. LetF be a regular net of S1 and a probability measure on S1. If is

supposed to be of full support, then, for any q R,

(q) := lim infr0

log

JFr (J)q

log r.

Proof. According to Definition 1.3, one has (q) = lim infr0 log Z(r)/log r, with

Z(r) := inf

i

(Bi )q; {Bi}i

,

the infimum being taken over the r-covers ofS1; for any r > 0, define

Z(r) :=

JFr(J)q .

Let mr be the integral part of 1/r and consider Br := {Bi}mri=0 where Bi := [ir,(i + 1)r],(0 i < mr ) and Bmr = [1 r, 1] (Br is an r/2-cover ofS1).

First, suppose that q < 0; by definition, one has |J| < r/2 for any J Fr/2 and thuseach ball Bi Br contains at least one interval in Fr/2, say Ji , so that

Z(r/2)

i

(Bi )q 2

JFr/2

(J)q 2Z(r/2)

(the factor 2 appears because one may have Jmr 1 = Jmr ). By the regularity ofF onealso has 2r |J|, for any J F2cr (where c > 1 stands for the constant involved in theregularity property ofF); hence, each J F2cr contains at least one Bi Br , implyingthat Z(2cr ) 2Z(r/2); one deduces that (q) is the lower limit of the ratio log Z(r)/log rwhen r tends to zero.

We now consider the case of q 0. By the regularity ofF, one has r |J| for anyJ Fcr ; hence, each Bi Br intersects no more than two elements ofFcr , one of them,say Ji , satisfying (Bi ) 2(Ji ). One may have Ji = Jj for i = j, but since each intervalin Fcr intersects at most N elements ofBr , for some N independent ofr , one gets

i

(Bi )q 2q N

i

(Ji )q 2qN

JFcr

(J)q, i.e.1

2qNZ(r/2) Z(cr). (45)

Suppose thatC is an arbitrary r-cover ofS1. It follows from Besicovitchs covering Lemma

(cf. [28, p. 30]) that there exists a sub-cover C ofC, such that each point x S1 belongsto at most three balls in C. One can check that, for 0 i mr , the number of C Cwhich intersects Bi is bounded by 6. Using again the fact that J Fcr intersects at mostN elements in Br , it is clear that J intersects at most 6N elements ofC; since any ball inC intersects no more than three elements ofFcr ,

JFcr(J)q 3(6N)q

CC

(C)q 3(6N)qCC

(C)q .

Taking the infimum over the r-cover C leads to Z(cr) 2(6N)q Z(r). With (45), oneconcludes that (q) is the lower limit of the ratio log Z(r)/log r when r tends to zero. 2

We now turn to the proof of Theorem C itself, which we split into two steps (the first

one being inspired by an argument in [36]).



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First step. To begin with, suppose that is a Gibbs measure of a Holder continuous

potential (which, according to our definition, implies that P() = 0). For any q Rand any d, the trivial identity

exp(Snq ()) =exp(qSn())

exp( (q)Sn0())(46)

holds for q := q (q)0. Denote by q the unique T-ergodic Gibbs measureassociated to the Holder continuous potential q ; since and the Lebesgue measure are

respectively Gibbs measures of and 0, the identity (46) gives, for any word w,

q [w]/R [w]q

|[w]| (q) Rq[w] (47)

for some constant R > 1. The Markovian net M being regular, it is possible by

Theorem 3.10 to consider the Lq -spectrum defined by the mean of the Moran partitionsMr (0 < r < 1). By a summation of (47) over the [w] Mr , there exists a constantR > 1 such that

1/R Z(r, q)/r (q) R, where Z(r, q) =

JMr(J)q;

taking the limit when r tends to 0, one concludes that (q) = (q), for any q R.

Second step. We now turn to the general case. Let us consider a sequence of Holder

continuous potentials k which are uniformly convergent to . Since P() = 0, one has|P (k )| k and thus (k P (k )) 2k ; hence one canassume that P (k )

=0. We denote by k the unique -ergodic Gibbs measure of k; by

the first step of the proof, k (q) = k (q), for any q R (k denotes the Lq -spectrumof k). Moreover, it follows from the classical properties of the topological pressure that

k tends to uniformly on the compact intervals. It remains to prove the pointwise

convergence ofk (q) to (q). Given > 0, one has k whenever n is largeenough; therefore, for any integer m > 0 and any word w of length m, one has

[w] K(m)Ck exp(2m)k[w] (48)where m K(m) (respectively Ck) is the sub-exponential sequence (respectivelyconstant) which characterizes the weak Gibbs property of (respectively the Gibbs

property of k , for k 1). Let Nr be the maximal length of the words w such that[w] Mr ; for any k 1 we consider the partition function Zk (r, q) := JMr k (J )q ,so that from (48)

Z(r,q) K(Nr )Ck exp(2Nr )Zk(r, q),or equivalently

1

log rlog Z(r, q) 1

log rlog Zk (r, q) +

Nr

log r

log K(Nr )

Nr+ Ck

log r+ 2Nr

log r

.

When r tends to 0, one gets (q) k (q) 2a , where a is a constant such thatNr /log(1/r ) a for any 0 < r < 1. The symmetric argument yields (q) k (q) + 2aand the proof of Theorem C is complete.



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Acknowledgements. De-Jun Feng was supported by an HK RGC grant and the Special

Funds for Major State Basic Research Projects and Eric Olivier was also supported byan HK RGC grant. The second author would like to thank N. Sidorov for pointing out

the problem of the Gibbs properties of the Erdos measure and specially A. Thomas for

enlightening and substantial discussions on this subject; he is also grateful to A. H. Fan,

J. P. Kahane, F. Ledrappier, P. Liardet, J. Peyriere and J. P. Thouvenot. The paper was

written during a postdoctoral visit to the Chinese University of Hong Kong and both

authors would like to thank K. S. Lau for his kind invitation and valuable discussions.

A. Appendix. First-order phase transition

It was first proved in [13] that the Lq -spectrum of the Erdos measure is not differentiable

at the critical value qc defined in Theorem 2.10; the present appendix is devoted to a self-

contained proof of this result.

THEOREM A.1. The Lq -spectrum is not differentiable at the critical value qc.

Any word w {0, 1, 2}n is associated to a unique sequence of (possibly empty) wordsm1, . . . , mk (1 k n) on the alphabet {0, 2} such that w1 = m11 mk1; from (29) onehas [w1] = [m11] [mk1]; since |[w1]| = |[m11] | | [mk1]|, one deduces that

[m11 mk1]q|[m11 mk1]|t

=k

i=1( tVPmi V )

q ( t/2q )2|mi |+3. (49)

Recall that

P (q,t)=

limn

1

nlog

Zn(q,t) where

Zn(q,t)

:= w{0,1,2}n[w1]q

|[w1]|t,

so that, with zn(q) :=

w{0,2}n (tVPwV )

q , one gets

Zn(q,t) =n

k=1

a1,...,ak0

a1++ak=n+1k

ki=1

zai (q)(t/2q )2ai+3. (50)

PROPOSITION A.2.

(i) qc < 2.25 and(ii)

n=0 nzn(q) < for any q < 2.25.

We shall give the proof of Proposition A.2 at the end of the present appendix.

PROPOSITION A.3. The following three propositions hold:

(i) X(q) := (q)/2q = sup 0 x 1;i=0 zi (q)x2i+3 1;(ii) q qc

i=0 zi (q) 1;

(iii) qc q < 2.25

n=0 zn(q)X(q)2n+3 = 1.

Proof. (i) For x > 0, define tq (x) := q log2/log + log x/log so that, for any q R,X(q) = max{x;P(q,tq (x)) 0} = min{x;P(q,tq (x)) 0}.

By definition {0, 2}0 = {} and P is the identity matrix, so that z0(q) = ( tVPV )q = 2q .



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For any (q,x)

R

[0, 1

]we set F (q , x )

:= n zn(q)x2n+3 (

[0,

+]) and we

consider, for any q R,x(q) := sup{x 0; F (q , x ) 1} = sup{x 0; F(q,x) < 1}, (51)

the second equality being justified by Abels theorem, for x(q) is bounded by the radius of

convergence of the power series

i zi (q)x2i+3. It follows from (50) that

Zn(q,tq (x)) =n

k=1

a1,...,ak0

a1++ak=n+1k

ki=1

zai (q)x2ai+3

k=0

n=0

zn(q)x2n+3

k,

that is,

Zn(q,tq (x))

k=0(F(q,x))k

. (52)

Given any x 0 with F (q , x ) < 1, the upper bound in (52) implies that Zn(q,tq(x))is bounded and thus P (q,tq (x)) 0: this means that x X(q), and from the secondequality in (51), one deduces that x(q) X(q).

For the converse inequality, let n 1 and r 1 be given so thatnr

m=1Zm(q,t)

m1,...,mn{0,2}|m1|,...,|mn| 0 such that z0 :=r01

i=0 zi (q)x2i+3 > 1; from (53), one deduces

that

limn

1

nlog

nr0k=1

Zk(q,tq (x)) log z0 > 0.

This is inconsistent with the fact that Zn(q,tq (x)) en for > 0 and any n largeenough and thus P(q,tq (x)) 0 and X(q) x: one concludes thatX(q) x(q), for xcan be taken arbitrarily close to x(q). Finally, since (q) q log2/log , for any q R,one necessarily has 0 X(q) 1, which complete the proof of (i).

(ii) According to Theorem 2.10, one has q qc if and only if (q)/2q = 1, whichleads to (ii) as a straightforward consequence of (i).

(iii) Notice that the map F : (q,x) F (q , x ) is finite and continuous at any point(q,x) ], 2.25[ [0, 1], for part (ii) of Proposition A.2 ensures that

0 F (q , x )

n

zn(q) < ,



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whenever q 1 for any qc < q , which

implies that F (qc, 1) 1, by the fact that F is continuous at (qc, 1): since (ii) also impliesthat F (qc, 1) 1, one concludes that F (qc, 1) = 1, completing the proof of (iii) and ofthe proposition as well. 2

Proof of Theorem A.1. We shall prove that X is not differentiable at qc; since by

Proposition A.3, one has X(q)

=1 whenever q

qc, this will be established if one shows

that X(q+c ) < 0. According to part (iii) of Proposition A.3, for any qc q < 2.25, onehas the equation

n=0

zn(qc)X(qc)2n+3 =

n=0

zn(q)X(q)2n+3,

or equivalently, using the fact that X(qc) = 1,

n=0(zn(qc) zn(q)) =

n=0

zn(q)(X(q)2n+3 1)

= (X(q) X(qc))

n=0zn(q)

2n+2i=0

X(q) i .

Finally one can write

X(q) X(qc)q qc

=

n=0 (zn(q) zn(qc))/(q qc)n=0 zn(q)

2n+2i=0 X(q)i

. (54)

If q tends to qc with qc < q < 2.25, then2n+2

i=0 X(q)i tends to 2n + 3 and since by

Proposition A.2 one has

n=0 zn(qc) (2n + 3) < , one concludes from (54) that

X(q+c ) =

n=0

w{0,2}n (

tVPwV )qc log( tVPwV )

n=0 zn(qc)(2n + 3)< 0. 2

Proof of Proposition A.2. Given (an)n=1 a sequence of positive integers, it is easily

checked by induction that, for any k

1,

tVQa1 Qak V (1 + a1) (1 + ak1)(2 + ak); (55)likewise, ifA(a1, . . . , a2k ) := (1+ a1a2)(1+ a3a4) (1+a2k1a2k) then, for = 0 or 1,

tVQa1 Qa2k+ V

A(a1, . . . , a2k) if = 0,A(a1, . . . , a2k)(1 + a2k+1) if = 1.

(56)

In what follows, will stand for the Riemann zeta function (i.e. (x) = n=1 1/nx ,x > 1).



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(i) When q 1, proving that qc < 2.25.

(ii) Let q < 0; from (56) and the fact that (1 + a2k+1)q

aq

2k+1, one gets

n=0nzn(q) 2

n=0

n(2 + n)q + 2

k=1

a1,...,a2k1

(a1 + + a2k)A(a1, . . . , a2k)q

+ 2

k=0

a1,...,a2k+11

(a1 + + a2k+1)A(a1, . . . , a2k)q aq2k+1

= 2

n=0n(2 + n)q + 2

k=1

a1,...,a2k1

2ka1A(a1, . . . , a2k)q

+ 2

k=1 a1,...,a2k+112ka1A(a1, . . . , a2k)

q aq2k+1

+ 2

k=1

a1,...,a2k+11

a2k+1A(a1, . . . , a2k)q aq2k+1

= 2

n=0n(2 + n)q + 4

n,m1

n(1 + nm)q

k=1k

n,m1

(1 + nm)qk1

+ 4

n,m1n(1 + nm)q

n=1

nq

k=1k

n,m1

(1 + nm)qk1

+ 2

n=1nq+1

k=1

n,m1

(1 + nm)qk

.

On the one hand, for any q < 2 one has n=0 n(2 + n)q < and

n,m1n(1 + nm)q 2

n=1

n(1 + n)q +

n=2nnq

n=2nq

< . (57)

Thus, there exist three positive constants C, C and C such that

n=0

nzn(q) C + C

k=1k

n,m1

(1 + nm)qk1

+ C

k=1

n,m1

(1 + nm)qk

. (58)



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On the other hand, one has for any q 1

(1 + nm)q + 2

n=1(1 + n)q 2q

n,m>1

(nm)q + 2

n=2nq 2q

=

n=2nq2

+ 2

n=2nq 2q = ( (q) 1)2 + 2( (q) 1) 2q .

For 0 > 0 such that 20 + 20 1/8 = 1, one can check that (2.25) 1 < 0, so that

nm1

(1

+nm)2.25 < 20

+20

1/8

=1. (59)

Therefore, by (58) and (59), one concludes that

n nzn(q) < when q < 2.25. 2

B. Appendix. Box and packing dimensions

Let M be a subset ofS1. An -packing ofM (0 < < 1) is a collection {Ji}i of mutuallydisjoint intervals Ji Fwith |Ji | and which intersect M; for any 0 1 we set

P (M) := lim0

sup

i

|Ji |; {Ji}i

,

where {Ji}i runs over the -packing of M (P is not sigma-additive). The box dimensionofM is by definition

dimB M := inf{;P (M) = 0} = sup{;P(M) = };

in general, dimB

n Mn = supn dimB Mn, whenever M1, M2, . . . form a countable familyof subsets ofS1. There are many other equivalent definitions of the box dimension ofM,

one of them using a regular net of S1: suppose that F := n Fn is a net ofS1, then, forany 0 1, we set

P (M|F) := lim0

sup

i

|Ji |; {Ji}i

,

where {Ji}i runs over the -packing of M by intervals in F. Recall that F is said to beregular if there exists a constant c > 1 such that for any n one has

|J

|/

|J

| c, for any

J Fn, J Fn+1 and J J; ifF is regular thendimB M := inf{;P(M|F) = 0} = sup{;P (M|F) = }

and one recovers the usual definition of the box dimension by using the dyadic net for

example. The notion of packing dimension has been introduced by Tricot in [44]; the

packing dimension ofM is

dimP M := inf

supn

dimB Mn; M

n

Mn

,



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with the important property that dimPn Mn = supn dimP Mn; moreover the packingdimension is a useful notion to get an upper bound of the Hausdorff dimension sincedimH M dimP M dimB M.

A systematic approach and detailed proof about fractal dimensions can be found in

[11, 28]; we also refer to [35] for a point of view related to dynamical systems.

REFERENCES

[1] P. Billingsley. Ergodic Theory and Information. John Wiley, New York, 1965.

[2] R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in

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Multifractal Analysis of Weak Gibbs Measure

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