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: dfeng@math.tsinghua.edu.cn) 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)
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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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1754 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1755
: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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Multifractal analysis of weak Gibbs measures 1757
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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1760 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1761
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
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Multifractal analysis of weak Gibbs measures 1763
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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1764 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1765
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).
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1766 D.-J. Feng and E. Olivier
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=: .
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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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1768 D.-J. Feng and E. Olivier
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.
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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.
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1770 D.-J. Feng and E. Olivier
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() ().
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Multifractal analysis of weak Gibbs measures 1771
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).
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1772 D.-J. Feng and E. Olivier
(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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Multifractal analysis of weak Gibbs measures 1773
(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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1774 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1775
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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1776 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1777
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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1778 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1779
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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1780 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1781
(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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1782 D.-J. Feng and E. Olivier
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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Multifractal analysis of weak Gibbs measures 1783
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
Mathematics, 470). Springer, Berlin, 1975.
[3] M. Bramson and S. A. Kalikow. Nonuniqueness in g-functions. Israel J. Math. 84 (1993), 153160.
[4] G. Brown, G. Michon and J. Peyriere. On the multifractal analysis of measures. J. Stat. Phys. 6 (1992),
775790.
[5] H. Cajar. Billingsley Dimension in Probability Spaces (Lecture Notes in Mathematics, 892). Springer,
Berlin, 1981.
[6] R. Cawley and R. D. Mauldin. Multifractal decomposition of Moran fractals. Adv. Math. 92 (1992), 196
236.
[7] P. Collet, J. L. Lebowitz and A. Porzio. The dimension spectrum of some dynamical systems. J. Stat.
Phys. 47 (1987), 609644.
[8] J. M. Dumont, N. Sidorov and A. Thomas. Number of representations related to a linear recurrent basis.
Acta Arith. 88 (1999), 391396.
[9] F. J. Dyson. Existence of phase-transition in a one-dimensional Ising ferromagnet. Comm. Math. Phys.
12 (1969), 91107.
[10] P. Erdos. On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 (1939), 974976.
[11] K. Falconer. Fractal GeometryMathematical Foundations and Applications. John Wiley, Chichester,
1990.
[12] A.-H Fan, K.-S Lau and S.-M Ngai. Iterated function systems with overlaps. Asian J. Math. 4 (2000),
527552.
[13] D.-J. Feng. The limit Rademacher functions and Bernoulli convolutions associated with Pisot numbers I,
II. Preprint, 1999.
[14] D.-J. Feng. Smoothness of Lq -spectrum of self-similar measures with overlaps. J. Lond. Math. Soc. 68
(2003), 102118.
[15] M. Fisher. The theory of condensation and the critical point. Physics 3 (1967), 255283.
[16] T. Halsey, M. Jensen, L. Kadanoff, I. Procaccia and B. Shraiman. Fractal measures and their singularities:
the characterization of strange sets. Phys. Rev. A 33 (1986), 11411151.
[17] Y. Heurteaux. Estimation de la dimension inferieure et de la dimension superieure des mesures. Ann. Inst.
Henri Poincare 34 (1998), 309338.
[18] F. Hofbauer. Examples for the nonuniqueness of the equilibrium state. Trans. A mer. Math. Soc. 228
(1977), 223241.
[19] F. Hofbauer. Local dimension for piecewise monotonic maps on the interval. Ergod. Th. & Dynam. Sys.
15 (1995), 11191142.
[20] T.-Y. Hu and K.-S. Lau. Multifractal structure of convolution of the cantor measure. Adv. App. Math. 22
(2000), 116.
[21] B. Jessen and A. Wintner. Distribution functions and the Riemann zeta function. Trans. Amer. Math. Soc.
38 (1935), 4888.
[22] M. Keane. Strongly mixing g-measures. Invent. Math. 16 (1972), 309324.
[23] A. Khintchin. Continued Fractions. University of Chicago Press, Chicago, 1964.
[24] K.-S. Lau and S.-M. Ngai. Lq -spectrum of the Bernoulli convolution associated with the golden ratio.
Stud. Math. 131 (1998), 1729.
[25] K.-S. Lau and S.-M. Ngai. Multifractal measures and a weak separation condition. Adv. Math. 141 (1999),
4596.
http://www.paper.edu.cn
-
7/27/2019 Multifractal Analysis of Weak Gibbs Measure
34/34
1784 D.-J. Feng and E. Olivier
[26] F. Ledrappier. Principe variationnel et systemes dynamiques symboliques. Z. Wahr. Verw. Geb. 30 (1974),
185202.[27] F. Ledrappier and A. Porzio. A dimension formula for Bernoulli convolutions. J. Stat. Phys. 76 (1994),
225251.
[28] P. Mattila. Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge,
1995.
[29] E. Olivier. Dimension de Billingsley des ensembles satures. C. R. Acad. Sci. Paris 328 (1998), 1316.
[30] E. Olivier. Multifractal analysis in symbolic dynamic and distribution of pointwise dimension for
g-measures. Nonlinearity 12 (1999), 15711585.
[31] E. Olivier. Structure multifractale dune dynamique non expansive definie sur un ensemble de Cantor.
C. R. Acad. Sci. Paris 331 (2000), 605610.
[32] E. Olivier, N. Sidorov and A. Thomas. On the Gibbs properties of some Bernoulli convolutions related to
-numeration. Preprint, 2002.
[33] L. Olsen. A multifractal formalism. Adv. in Math. 116 (1995), 82196.
[34] L. Olsen. Multifractal geometry. Progr. Probability 46 (1998), 338.
[35] Y. Pesin. Dimension Theory in Dynamical Systems. University of Chicago Press, Chicago, 1997.
[36] Y. Pesin and H. Weiss. The multifractal analysis of Gibbs measures: motivation, mathematical foundation,and examples. Chaos 7 (1997), 89106.
[37] J. Peyriere. Calcul de dimensions de Hausdorff. Duke Math. J. 44 (1976), 591601.
[38] H. Rao and Z. Y. Wen. A class of self-similar fractals with overlap structures. Adv. Appl. Math. 20 (1998),
5072.
[39] D. A. Rand. The singularity spectrum f() for cookie-cutters. Ergod. Th. & Dynam. Sys. 9 (1989), 527
541.
[40] D. Ruelle. Statistical Mechanics. Rigorous Results. Benjamin, New York, 1969.
[41] O. Sarig. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999),
15651594.
[42] N. Sidorov and A. Vershik. Ergodic properties of the Erdos measure, the entropy of the golden shift and
related problems. Monatshefte Math. 126 (1998), 215261.
[43] R. S. Strichartz, A. Taylor and T. Zhang. Density of self-similar measures. Exp. Math. 4 (1995), 101128.
[44] C. Tricot. Two definitions of fractional dimension. Math. Proc. Cambr. Philos. Soc. 91 (1982), 5774.
[45] P. Walters. An Introduction to Ergodic Theory. Springer, Berlin, 1982.
[46] M. Yuri. Zeta functions for certain non-hyperbolic systems and topological Markov approximations.Ergod. Th. & Dynam. Sys. 17 (1997), 9971000.
http://www.paper.edu.cn
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