On the Hausdorff dimension of invariant measures for multicritical circle maps
aa r X i v : . [ m a t h . D S ] O c t ON THE HAUSDORFF DIMENSION OF INVARIANTMEASURES FOR MULTICRITICAL CIRCLE MAPS
FRANK TRUJILLO
Abstract.
We give explicit bounds for the Hausdorff dimension of the uniqueinvariant measure of C multicritical circle maps without periodic points.These bounds depend only on the arithmetic properties of the rotation number. Introduction
The object of study in this work are C r multicritical circle maps of the circle S = R \ Z . A map f : S → S is called a C r multicritical circle map if it isan orientation preserving circle homeomorphism of class C r having finitely manycritical points c , . . . , c n − and taking the form x f ( c i ) + x | x | p i − in a suitable coordinate system around each critical point for some p i > . Thereal number p i is called the power-law exponent or criticality of the critical point c i . Let f : S → S be an orientation-preserving circle homeomorphism and let F : R → R be a lift of f , that is, a continuous homeomorphism of R such that F ( x + 1) = F ( x ) + 1 and F ( x )( mod 1) = f ( x ) for all x ∈ R . By a classical resultof Poincaré the limit ρ ( f ) = lim n →∞ F n ( x ) n mod 1 is well defined and independent of the value x ∈ R initially chosen. This limitis called the rotation number of f . We say that a multicritical circle map is irrational if its rotation number is irrational. Recall that the rotation numberof a circle homeomorphism is irrational if and only the map does not possessany periodic points. Furthermore, maps with irrational rotation number admit aunique invariant measure. For a simple proof of this fact we refer the reader to [6].It was proven by Khanin in [12] that the unique invariant measure µ of anyirrational multicritical circle map f is singular with respect to the Lebesguemeasure, that is, there exists a measurable set X ⊂ S of zero Lebesgue measurefor which µ ( X ) = 1 . An alternative proof of this fact was given in [7] by Graczykand Świątek. Since this unique invariant measure is singular, it is natural toinvestigate its Hausdorff dimension . The Hausdorff dimension of any probability
Borel measure on S is defined as dim H ( µ ) := inf { dim H ( X ) | µ ( X ) = 1 } , where dim H ( X ) denotes the Hausdorff dimension of the set X ⊂ S . We providethe formal definition of Hausdorff dimension in the next section.By a classical result of Yoccoz [19] any C irrational critical circle map f istopologically conjugated to the irrational rotation x x + ρ ( f ) . Hence any twoirrational multicritical circle maps with the same rotation number are conjugatedto each other by a C homeomorphism. It was proven in [4] by Estevez and deFaria that if these two maps have the same number of critical points any conjugacybetween them is also quasisymmetric. In the same work the authors conjecturethat for C irrational multicritical circle maps their differentiable conjugacy classdepends only on their signature . Given a multicritical circle map f with n f critical points c , . . . , c n f − and unique invariant measure µ f its signature is the (2 n f + 2) tuple ( n f , ρ ( f ); p , . . . , p n f − ; λ , . . . , λ n f − ) , where p i is the criticality of c i and λ i = µ f [ c i , c i +1 ] . Since the Hausdorff di-mension of a set is preserved by diffeomorphisms, the conjecture implies thatthe Hausdorff dimension of the unique invariant measure of C irrational multi-critical circle maps depends only on their signature. The previous conjecture isa generalization of the corresponding one for unicritical circle maps, where theonly invariants are the rotation number and the criticality of the unique criticalpoint. Guarino, Martens and De Melo [8] recently proved a weak version of thisconjecture for C irrational unicritical maps with the same odd criticality.The main result of this paper gives a relation between the Hausdorff dimensionfor the unique invariant measure of a C irrational multicritical circle map and thearithmetic properties of its rotation number. We will recover, as a by-product ofour constructions (Corollary 3.2), a proof of the singularity of the unique invariantmeasure of irrational multicritical maps, in the same spirit of that in [12] forthe unicritical case, for maps whose rotation number is not of bounded type.Before stating our main result let us introduce some notations that will be usedthroughout this work. A real number α is said to be Diophantine if there exist γ > and τ ≥ such that(1) (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ γq τ +2 for all p, q ∈ Z , q = 0 . Given γ > and τ ≥ we denote by D ( γ, τ ) the set of real numbers α verifying(1). A number in D ( γ, τ ) is called a Diophantine number of type ( γ, τ ) . Let D τ = [ γ> D ( γ, τ ) , D = [ τ ≥ D τ . AUSDORFF DIMENSION FOR MULTICRITICAL CIRCLE MAPS 3
Recall that for any τ > the set D τ has full Lebesgue measure while D has zeroLebesgue measure. A number in D is said to be of bounded type . We now statethe main result of this note. Theorem 1.1.
Let f be an irrational multicritical circle map with unique invari-ant measure µ . There exists a positive constant ν , depending only on ρ ( f ) , suchthat the following holds: • If ρ ( f ) ∈ D τ for some τ ≥ then dim H ( µ ) ≥ τ + ν . • If ρ ( f ) / ∈ D τ for some τ > then dim H ( µ ) ≤ τ + 1 . The upper and lower bounds in Theorem 1.1 will be proven in Propositions 3.3and 3.4 respectively.To end this section let us make a few comments about several related results.Graczyk and Świątek studied in [7] the Hausdorff dimension of the unique in-variant measure of multicritical circle maps with rotation number of boundedtype. For this particular class the authors show that the associated Hausdorffdimension is bounded away from and . Theorem 1.1 allows to recover thelower bound in [7] while providing an explicit estimate.For circle diffeomorphisms, it follows from the works of Herman [9] and Yoc-coz [18] that sufficiently regular circle diffeomorphisms are smoothly conjugatedto a rigid rotation provided its rotation number is Diophantine. Hence, for anysmooth circle diffeomorphism with Diophantine rotation number, its unique in-variant measure is equivalent to the Lebesgue measure and therefore its Hausdorffdimension is equal to one. On the other hand, for any ≤ β ≤ and any Liouville number α , that is, any non-Diophantine irrational number, Sadovskaya [15] hasconstructed, using the Anosov-Katok method [1], examples of smooth diffeomor-phisms with rotation number α whose unique invariant measure has Hausdorffdimension β .In the case of circle homeomorphisms with a break , i.e. smooth diffeomorphismswith a singular point where the derivative has a jump discontinuity, Khanin andKocić [11] have shown that for almost any irrational number α the unique invari-ant measure of a C ǫ circle homeomorphism with a break and rotation numberequal to α has zero Hausdorff dimension. FRANK TRUJILLO Preliminaries
Continued fractions.
We state some basic results on continued fractionsand Diophantine properties. For more details see [13]. Let α ∈ (0 , irrationaland denote by [ a , a , a , . . . ] its continued fraction expansion α = 1 a + 1 a + 1 a + 1 · · · = [ a , a , a , . . . ] . The n-th convergent of α is given by p n q n = [ a , a , a , . . . , a n ] and satisfies min ≤ p,q ≤ q n (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) = ( − n +1 (cid:18) α − p n q n (cid:19) < q n q n +1 . If we set q − = 0 , q = 1 the following recursive relation holds(2) q n = a n q n − + q n − , for n ≥ . The denominators q n are called the return times of α . From (2) iseasy to show that they grow exponentially fast, in fact q n > ( √ n for n ≥ . The Diophantine numbers can be characterized in terms of the continued fractionexpansion as follows. Given τ ≥ an irrational number α belongs to D τ if andonly if sup n q n +1 q τ +1 n < ∞ . The last inequality is equivalent to sup n a n +1 q τn < ∞ . Dynamical Partitions.
Let f be an orientation preserving circle homeo-morphism with irrational rotation number α = ρ ( f ) = [ a , a , . . . ] . Using thereturn times q n of α described in the previous section we define a partition of S as follows.Fix x ∈ S and let x i = f i ( x ) . Denote by I n the circle arc [ x , x q n ) if n iseven and [ x q n , x ) if n is odd. Let I in = f i (cid:0) I n (cid:1) . Then P n ( x ) = { I n − , I n − , . . . , I q n − n − } ∪ { I n , I n , . . . , I q n − − n } AUSDORFF DIMENSION FOR MULTICRITICAL CIRCLE MAPS 5 defines a partition of S . For every x ∈ S let P n ( x ) be the atom of P n containing x . The next relations follow directly from the continued fraction expansion prop-erties described in the previous section. For all ≤ i < q n we have I in +1 ⊂ I in − and(3) I in − \ I in +1 = a n +1 − [ j =0 I i + q n − + jq n n . Hence, it is clear that P n +1 ( x ) is a refinement of P n ( x ) where each I in − ∈ P n ( x ) is divided into a n +1 + 1 pieces of P n +1 ( x ) . The RHS of (3) is the union of a n +1 different iterates by f of I n , which are actually adjacent intervals in the circle.In the following we denote these adjacent intervals by ∆ ( n ) i,j = I i + q n − + jq n n , for all ≤ i < q n , ≤ j < a n +1 . To simplify the notation, for i = 0 , we writesimply ∆ ( n ) j instead of ∆ ( n )0 ,j . ∆ ( n )0 • ∆ ( n )1 • ∆ ( n )2 • ∆ ( n )3 • ∆ ( n )4 • ∆ ( n )5 • ∆ ( n )6 • ∆ ( n )7 •• • •• x q n x x q n +1 x q n − I n − I n +1 I n Figure 1.
Refinement of I n − for a rigid rotation with a n = 7 .Let f be a multicritical circle map with N different critical points. Let c bea critical point of f and consider the dynamical partition associated to it. Tosimplify the notation let us write P n instead of P n ( c ) . Let T n = f q n | I n − \ I n +1 . Denote r n = n < j < a n +1 (cid:12)(cid:12)(cid:12) ∆ ( n ) j contains a critical point of T n or j = a n +1 − o . Define k ( n )0 = 0 , k ( n ) r n = a n +1 − , and let < k ( n )1 < k ( n )2 < · · · < k ( n ) r n − < a n +1 − , be the indices of the intervals in { ∆ ( n ) j } Lebesgue measure of a set A ⊂ S by | A | . Givena positive constant M and two positive real numbers x, y we write x ≍ M y if theinequality M − y < x < M y is verified. We now summarise some of the geometric properties of dynamicalpartitions associated to critical points of multicritical circle maps. All of theproperties we state here, except for the first one which goes back to Herman [10]and Świątek [17], follow from the results of Estevez and de Faria in [4]. Theorem 2.1. There exists a positive constant M such that for any irrationalmulticritical circle map f of class C and for all n ≥ n , where n is a sufficientlylarge natural number depending only on f , the partition P n , associated to any ofits critical points, satisfies the following: (1) If I, J are any two adjacent atoms of P n then | I | ≍ M | J | . (2) For each non-empty bridge G ( n ) i,s | I in − | ≍ M (cid:12)(cid:12) G ( n ) i,s (cid:12)(cid:12) . AUSDORFF DIMENSION FOR MULTICRITICAL CIRCLE MAPS 7 (3) For all ≤ i < q n and all ≤ s ≤ r n | I in − | ≍ M (cid:12)(cid:12)(cid:12) ∆ ( n ) i,k ( n ) s (cid:12)(cid:12)(cid:12) . (4) For all ≤ i < q n , all ≤ s ≤ r n and all k ( n ) s < j < k ( n +1) s (6) (cid:12)(cid:12) ∆ ( n ) i,j (cid:12)(cid:12) ≍ M | I in − | min { j − k ( n ) s , k ( n +1) s − j } . (5) If ρ ( f ) = [ a , a , . . . ] (7) min I ∈P n | I | ≥ M − n ( a a · · · a n ) . •••• • •• ••• • •• ••• x q n x x q n +1 x q n − ∆ ( n ) k ( n )2 ∆ ( n ) k ( n )1 ∆ ( n ) k ( n )0 • •• •• • •• I n − I n +1 I n Figure 2. Refinement of I n − for a multicritical circle map with3 different critical times.Let us make a few comments about Theorem 2.1. We stress the fact thatthe constant M in the Theorem is universal , that is, independent of the map f . Estimates of this type are sometimes called beau after the seminal works ofSullivan [16].The first assertion is a classical result in the theory of critical circle home-omorphisms and, as we mentioned before, was initially proved by Herman [10]and Świątek [17]. It states the bounded geometry of the system, that is, the factthat any two neighbouring atoms of the dynamical partitions have comparablesize. This appears in sharp contrast to the case of circle diffeomorphisms wherethe length ratio of the neighbouring intervals I n − , I n in the dynamical partitioncan be arbitrarily small. In fact, for an irrational rigid rotation x x + α with α = [ a , a , . . . ] , the dynamical partition associated to any point in S satisfies (cid:12)(cid:12) I n − (cid:12)(cid:12)(cid:12)(cid:12) I n (cid:12)(cid:12) > a n +1 . A recent proof of the first assertion can be found in [4]. The second and thirdclaims are proven in Proposition 4.1 of [4]. These assertions can be regarded asthe generalization of the following known fact about the geometry of dynamicalpartitions of multicritical circle maps, which we quote from [7]: For any elementof any dynamical partition, the ratios of its length to the length of the extreme FRANK TRUJILLO intervals of the next partition subdividing it are bounded by a uniform constant.The last two properties, although not explicitly stated in [4], follow directlyfrom the results therein proved. Notice that the last claim is a consequence of thefourth assertion, up to increase the constant M if necessary, by a simple inductiveargument. In the unicritical case this was established by de Faria and de Meloin [2].Particular instances of the fourth assertion are used in [4], specially in the con-struction of a balanced decomposition for the bridges in the dynamical partition.For completeness and for the convenience of the reader, let us sketch the proofof this assertion. We start by introducing the main tools appearing in the proof,which, although will not be explicitly used in latter sections, are implicitly at thecore of this work. Schwarzian derivative. For a C map f defined on an interval or S its Schwarzian derivative at any non-critical point is given by: Sf ( x ) = D f ( x ) Df ( x ) − (cid:18) D f ( x ) Df ( x ) (cid:19) . Almost parabolic maps. Let a ≥ and I , ..., I a +1 be adjacent intervals onthe circle. A negative Schwarzian derivative map f : I ∪ · · · ∪ I a → I ∪ · · · ∪ I a +1 such that f ( I i ) = I i +1 is called an almost parabolic map . The intervals I , ..., I a +1 are called the fundamental domains of f . The following geometric estimate isdue to Yoccoz. See [2] appendix B for a proof. Lemma 2.2. (Yoccoz’s Lemma) Let f : I = I ∪ · · · ∪ I a → I ∪ · · · ∪ I a +1 bean almost parabolic map and let σ > obeying | I | , | I a | ≥ σ | I | . There exists a positive constant C , depending only on σ , such that C − | I | min { j, a − j } ≤ | I j | ≤ C | I | min { j, a − j } for all < j < a. Distortion estimates. Given f : S → S and two intervals M ⊂ T ⊂ S wedefine the space of M inside T as s ( M, T ) = min (cid:26) | L || M | , | R || M | (cid:27) where L and R are the left and right components of T \ M . A proof of the nextProposition can be found in [3] (Section IV, Theorem 3.1). AUSDORFF DIMENSION FOR MULTICRITICAL CIRCLE MAPS 9 Proposition 2.3. (Koebe’s nonlinearity principle for real maps.) Let f : S → S of class C and let τ, l be positive constants. There exists K ( τ, l, f ) with the following property: If T is an interval such that f i | T is a diffeomorphismand if P i − j =0 | f j ( T ) | < l then for each interval M ⊂ T obeying s ( M, T ) ≥ τ andall x, y ∈ M K ≤ Df i ( x ) Df i ( y ) ≤ K. We have now all the tools to sketch the proof of the fourth assertion in Theorem2.1. Proposition 4.2 in [4] guarantees that for n sufficiently large and for criticaltimes k ( n ) s , k ( n ) s +1 satisfying k ( n ) s +1 − k ( n ) s > the map f q n | B ( n ) s : k ( n ) s +1 − [ j = k ( n ) s +2 ∆ ( n ) j → k ( n ) s +1 − [ j = k ( n ) s +3 ∆ ( n ) j is almost parabolic. The set B ( n ) s = k ( n ) s +1 − [ j = k ( n ) s +2 ∆ ( n ) j ⊂ G ( n ) s is called the s -th reduced bridge . Since f q n | B ( n ) s is almost parabolic, Yoccoz’sLemma together with 1 and 2 of Theorem 2.1 imply equation (6) for i = 0 . Theextension to other atoms of the partition is a consequence of Koebe’s nonlinearityprinciple. The fact that the constant in the Theorem can be taken to be univer-sal is a consequence of the beau bounds for cross-ratio inequalities of irrationalmulticritical maps proven by Estevez and de Faria in [5].2.3. Hausdorff dimension. For a subset X of a metric space M we define its d -dimensional Hausdorff content by C dH ( X ) := lim ǫ → inf ( U i ) X i ( diam ( U i )) d , where the infimum is taken over all countable covers ( U i ) of X satisfying diam ( U i ) <ǫ . The Hausdorff dimension of X is given by dim H ( X ) := inf { d ≥ | C dH ( X ) = 0 } . We recall that the Hausdorff dimension of a probability measure µ over M isgiven by dim H ( µ ) := inf { dim H ( X ) | µ ( X ) = 1 } . A proof of the following result can be found in [14]. Proposition 2.4. (Frostman’s Lemma) . Suppose that µ is a probability Borelmeasure on the interval and that for µ -a.e. point δ ≤ lim inf ǫ → log µ ( x − ǫ, x + ǫ )log ǫ ≤ δ . Then δ ≤ dim H ( µ ) ≤ δ . Proof of Theorem 1.1 In this section f will denote a C irrational multicritical circle map with N different critical points, rotation number α and unique invariant measure µ . Wefix one of its critical points and denote by {P n } n ∈ N its associated dynamicalpartitions. We will use the notations introduced in the preliminaries to denotethe continued fraction and return times of α .3.1. Upper bound. We will bound from above the Hausdorff dimension of µ by constructing appropriate full µ -measure sets whose Hausdorff dimension wecan control. Before giving their explicit definition let us first try to motivate theconstruction.Notice that for a n +1 sufficiently large the union of the “big” atoms of the par-tition P n , namely the union of the intervals I n − , I in − , . . . , I q n − n − , has µ -measureclose to . Indeed, by definition of the dynamical partitions(8) q n +1 δ n + q n δ n +1 , for all n ≥ , where δ n = µ ( I n ) . By (8)(9) δ n ≤ q n +1 . From (2) and (9)(10) q n − δ n ≤ a n +1 a n . Hence, by (8) and (10) µ q n − [ i =0 I jn − ! = q n δ n − ≥ − a n a n +1 . Moreover µ q n − [ i =0 I jn − \ I jn +1 ! = q n ( δ n − − δ n +1 ) ≥ − a n +1 . AUSDORFF DIMENSION FOR MULTICRITICAL CIRCLE MAPS 11 Let us recall equation (5), namely I in − \ I in +1 = r n [ s =0 ∆ ( n ) i,k ( n ) s ∪ G ( n ) i,s , where k ( n )0 , k ( n )1 , . . . , k ( n ) r n are the critical times associated to f q n | I n − \ I n +1 and ∆ ( n ) i,k ( n ) s , G ( n ) i,s are, respectively, the iterates of the critical spots and primary bridgesdefined in section 2.2. Recall also that the secondary bridge G ( n ) i,s is the union of k ( n ) s +1 − k ( n ) s − adjacent intervals G ( n ) i,s = k s +1 − [ j = k ( n ) s +1 ∆ ( n ) i,j . By (6) in Theorem 2.1 and provided k ( n +1) s − k ( n ) s is sufficiently big, the “centralintervals” of the secondary bridge G ( n ) i,s will have a much smaller Lebesgue mea-sure when compared to the first and last intervals. Nevertheless, as they are alliterates of I n , their measure with respect to µ must be the same. Having thisin mind we construct an appropriate full measure set to bound the Hausdorffdimension of the invariant measure µ .Given < γ < , denote by G ( n ) s,γ the union of the k ( n ) s +1 − k ( n ) s − ⌊ a γn +1 ⌋ centralintervals of G ( n ) s when seen as the union of adjacent atoms in P n +1 , namely G ( n ) s,γ = [ n ∆ ( n ) j (cid:12)(cid:12)(cid:12) k ( n ) s + a γn +1 < j < k ( n +1) s + 1 − a γn +1 o . Let(11) A ni,γ = r n [ s =0 f i (cid:0) G ( n ) s,γ (cid:1) , A nγ = q n − [ i =0 A ni,γ , for all ≤ i < q n . We have the following. Lemma 3.1. Let M, n as in Theorem 2.1 when applied to f . For any < γ < and all n ≥ n the following holds: (1) µ ( A ni,γ ) ≥ δ n ( a n +1 − (4 N + 2) a γn +1 ) . (2) µ ( A nγ ) ≥ (cid:16) − a n +1 (cid:17) (cid:16) − N +2 a − γn +1 (cid:17) . (3) | A ni,γ | ≤ (4 N +2) M | I in − |⌊ a γn +1 ⌋ . (4) | A nγ | ≤ (4 N +2) M ⌊ a γn +1 ⌋ . Proof. By definition of A ni,γ and the invariance of µ by f we have µ ( A ni,γ ) ≥ δ n r n X s =0 (cid:0) k ( n ) s +1 − k ( n ) s − a γn +1 (cid:1) ≥ δ n ( a n +1 − (4 N + 2) a γn +1 ) which proves the first assertion. By (8) and (10) µ ( A nγ ) = q n µ ( A n ,γ ) ≥ q n δ n a n +1 − N + 2 a − γn +1 ! = (1 − q n δ n +1 − q n − δ n ) − N + 2 a − γn +1 ! ≥ (cid:18) − a n +1 (cid:19) − N + 2 a − γn +1 ! . This proves the second claim. By (6) | A ni,γ | = X s ∈ K ( n ) γ k ( n ) s +1 −⌈ a γn +1 ⌉ X j = k ( n ) s + ⌊ a γn ⌋ +1 | ∆ ( n ) j |≤ M r n X s =0 | I in − | + ∞ X j = ⌊ a γn ⌋ +1 j = (4 N + 2) M | I in − |⌊ a γn +1 ⌋ which proves the third assertion. Since the intervals I n − , I n − , . . . , I q n − n − aredisjoint the last assertion follows directly from the third one. (cid:3) Corollary 3.2. Suppose ρ ( f ) is not of bounded type. Then the unique invariantmeasure of f is singular with respect to the Lebesgue measure.Proof. This follows directly from and of the previous Lemma. (cid:3) Proposition 3.3. Suppose ρ ( f ) / ∈ D τ for some τ > . Then dim H ( µ ) ≤ τ + 1 . Proof. Since ρ ( f ) / ∈ D τ there exists an increasing sequence ( n k ) k ∈ N of naturalnumbers obeying a n k +1 ≥ q τn k AUSDORFF DIMENSION FOR MULTICRITICAL CIRCLE MAPS 13 for all k ∈ N . Given < γ < , let A ni,γ , A nγ as in (11) and define A γ = \ i ≥ [ k ≥ i A n k γ . By Lemma 3.1 µ ( A n k γ ) −−−→ k →∞ . Thus µ ( A γ ) = 1 . By definition of Hausdorff dimension dim H ( µ ) ≤ inf <γ< dim H ( A γ ) . Let us show that dim H ( A γ ) ≤ τ +1 for γ sufficiently close to . Let d > τ +1 , ǫ > and take K sufficiently large so that diam ( P n K ) < ǫ . Thus C = (cid:8) A n k i,γ | K < k, ≤ i < q n k (cid:9) is an open cover of A γ with diameter less than ǫ . Hence C dH ( A γ ) ≤ lim inf K →∞ X k>K q nk − X i =0 | A n k i,γ | d ≤ lim inf K →∞ X k>K M d (4 N + 2) d ⌊ a γn k +1 ⌋ d q nk − X i =0 | I in k − | d ≤ lim inf K →∞ X k>K M d (4 N + 2) d a dγn k +1 q − dn k ≤ lim inf K →∞ X k>K M d (4 N + 2) d q − d ( γτ +1) n k . By hypothesis < d ( τ + 1) . Since the return times q n grow at least exponentiallythe sum in the last inequality converges for γ sufficiently close to 1. Thus C dH ( A γ ) = 0 for γ sufficiently close to . Therefore inf <γ< dim H ( A γ ) ≤ τ + 1 which finishes the proof. (cid:3) Lower bound. The lower bound for the Hausdorff dimension of the uniqueinvariant measure of f will be a direct application of Frostman’s Lemma (Propo-sition 2.4). Proposition 3.4. Suppose ρ ( f ) ∈ D τ for some τ ≥ . Let M > be as inTheorem 2.1. Then dim H ( µ ) ≥ τ + ν + ν log M , where ν = lim sup n →∞ a a . . . a n )log q n , ν = lim sup n →∞ n log q n . Proof. By Theorem 2.1 there exists a natural number n such that min ∆ ∈P n +1 | ∆ | > M − ( n +1) ( a · · · a n +1 ) , for all n ≥ n . Let x ∈ S such that x is not an end point of any of the intervalsof the dynamical partitions {P n } n ∈ N . Let Γ = sup n ∈ N a n +1 q τn . Notice that Γ < + ∞ since ρ ( f ) ∈ D τ . Let < ǫ < and define n ( x, ǫ ) = min { k ∈ N | ∃ ∆ ∈ P k +1 s.t. ∆ ⊂ B ǫ ( x ) } . Let ǫ sufficiently small so that n = n ( x, ǫ ) > n . Notice that B ǫ ( x ) must becontained in the union of two adjacent elements ∆ , ∆ ∈ P n . Let ∆ ∈ P n +1 such that ∆ ⊂ B ǫ ( x ) . Hence ǫ ≥ | ∆ | , µ ( B ǫ ( x )) ≤ µ (∆ ( n − ) ≤ q − n , which yields to log µ ( B ǫ ( x ))log ǫ ≥ log 2 µ (∆ ( n − )log | ∆ |≥ log q n − log 2log( a a . . . a n +1 ) + log M n +1 ≥ log q n − log 22 τ log(Γ q n ) + 2 log( a a ...a n ) + ( n + 1) log M . Therefore lim inf ǫ → log µ ( B ǫ ( x ))log ǫ ≥ τ + ν + ν log M . The result follows by Frostman’s Lemma. 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Comptes Rendus desSéances de l’Académie des Sciences. Série I. Mathématique 298 , 7 (1984), 141–144. IMJ-PRG, UP7D, 58-56 Avenue de France, 75205 Paris Cedex 13, France E-mail address ::