Dimension of measures: the probabilistic approach

Publ. Mat. 51 (2007), 243–290 DIMENSION OF MEASURES: THE PROBABILISTIC APPROACH Yanick Heurteaux Abstract Various tools can be used to calculate or estimate the dimension of measures. Using a probabilistic interpretation, we propose very simple proofs for the main inequalities related to this notion. We also discuss the case of quasi-Bernoulli measures and point out the deep link existing between the calculation of the dimension of auxiliary measures and the multifractal analysis. The notion of dimension is an important tool to classify the subsets in Rdand in particular to compare the size of small sets. There exist various definitions of dimension. The Hausdorff and the packing dimensions are probably the most famous one and can be considered as “extremal” notions of dimension. We refer to [Fal90] for precise definitions and we denote Hs(resp. ˆ Ps) the Hausdorff (resp. packing) measures. Finally, dim(E) and Dim(E) are respectively the Hausdorff and the packing dimension of a set E. The computation of the dimension of a set Eis naturally connected to the analysis of auxilliary Borel measures. The first elementary result in this direction is the following. Proposition 0.1. Let Ebe a Borel subset in Rdand mbe a Borel measure such that m(E)>0. Suppose that there exist s > 0and C > 0 such that ∀x∈E, m(B(x, r)) ≤Crsif ris small enough. Then, Hs(E)>0and dim(E)≥s. There is a converse to Proposition 0.1 known as Frostman’s Lemma (see for example [Mat95]). 2000 Mathematics Subject Classification. Primary: 28A12, 28A78, 60F10; Secondary: 28D05, 28D20, 60F20. Key words. Hausdorff dimension, packing dimension, lower and upper dimension of a measure, multifractal analysis, quasi-Bernoulli measures. 244 Y. Heurteaux Proposition 0.2. Suppose that Eis a Borel subset in Rdsuch that Hs(E)>0. There exists a Borel measure msuch that m(E)>0satisfying ∀x∈E, ∀r > 0, m(B(x, r)) ≤Crs. In particular, the result is true if dim(E)> s. Similar results, involving the packing dimension of the set Eare also true (see [Fal97, Propositions 2.2, 2.3 and 2.4]). In vue of Propositions 0.1 and 0.2, it is natural to introduce the local dimensions (also called H¨older exponents) of the measure mwhich are defined as dim m(x) = lim inf r→0 log(m(B(x, r))) log r dim m(x) = lim sup r→0 log(m(B(x, r))) log r. The quantities dim and dim are respectively called the lower and the upper local dimension of the measure mat point x. Finally, Propositions 0.1 and 0.2 can be reformulated as Proposition 0.3. Let Ebe a Borel subset in Rd. dim(E) = sup {s, ∃m, m(E)>0and dim m(x)≥s, ∀x∈E}. We can also refer to Tricot ([Tri82]) and Cutler ([Cut95]) who studied the link between the Hausdorff dimension (or the packing dimension) of a set Eand the local exponents of auxiliary measures. The deep relation between the value of the local exponent of auxiliary measures and the dimension of a given set Eis very useful in practice. In many situations, this is the natural way to compute the dimension of the set E. It is for example the case for self-similar sets. Let S1,...,Skbe similarities in Rdwith ratio 0 < ri<1 and Ebe the unique nonempty compact set such that E=S i Si(E) (see [Hut81]). For the sake of simplicity, suppose that the compact sets Si(E) are disjoint. Then Eis a Cantor set and the application (1) i= (i1,...,in,...)∈ {1,...,k}N∗7−→ \ n Si1◦ · · · ◦ Sin(E) is an homeomorphism. Let sbe the unique positive real number such that Pk i=1 rs i= 1 and mbe the unique probability measure such that m(Si1◦ · · · ◦ Sin(E)) = rs i1···rs in. On Dimension of Measures 245 The measure mis nothing else but the image of a multinomial measure on the symbolic Cantor set {1,...,k}N∗through the application (1). Computing the local exponents of m, we find dim(E) = Dim(E) = s. This result remains true if the so called Open Set Condition is satisfied (see [Hut81], [Fal97]). The case of self-affine sets is much more difficult ([McM84], [Urb90], [Ols98]). The thermodynamic formalism is an interesting tool to give the value of the Hausdorff dimension of sets that are obtained in more general dynamical contexts. This is for example the case for cookie-cutter sets ([Bed86], [Bed91]), graph-directed sets ([MW88]) and Julia sets ([Rue82], [Zin97]). We can also refer to [Fal97]. Another famous result, due to Eggleston ([Egg49]) concerns the occurence of digits in the ℓ-adic decomposition of real numbers. Let ℓ≥2, p= (p0,...,pℓ−1) a probability vector and x=P+∞ k=1 xkℓ−k∈[0,1) be the (proper) decomposition of the real number xin base ℓ. Finally let fi n(x) = 1 n♯{k∈ {1,...,n};xk=i} be the frequency of the digit i. If E(p) is the set of real numbers x∈(0,1) such that for all i∈ {0,...,ℓ−1}, lim n→+∞fi n(x) = pi, then (2) dim(E(p)) = Dim(E(p)) = − ℓ−1 X i=0 pilogℓpi. The proof of this result is based on the analysis of an auxiliary measure m defined by m " n X i=1 εiℓ−i, n X i=1 εiℓ−i+ℓ−n!!= n Y i=1 pεi. The strong law of large numbers easily ensures that the measure mis carried by the set E(p) and that dim m(x) = dim m(x) = − ℓ−1 X i=0 pilogℓpiif x∈E(p). Formula (2) follows (see Part 1 of the present paper for a detailed study of the case ℓ= 2). We can also reverse the point of view and try, for a given measure m in Rd, to compute or to estimate the dimension of sets that are naturally 246 Y. Heurteaux related to the measure m. In that way, we can in particular think to the negligible sets and the sets of full measure and define the quantities dim∗(m) = inf(dim(E); m(E)>0) dim∗(m) = inf(dim(E); m(E) = 1).(3) Dimension dim∗(m) first appears in [You82]. These two dimensions are respectively called the lower and the upper dimension of the measure m(see for example [Fal97] or [Edg98]). They precise how much the measure mis a “singular measure” or a “regular measure” and they are important quantities for the understanding of m. Similar definitions involving the packing dimension can also be proposed: Dim∗(m) = inf(Dim(E); m(E)>0) Dim∗(m) = inf(Dim(E); m(E) = 1).(4) There are numerous works in which estimates of the dimension of a given measure are obtained. In particular, a lot of papers deal with the harmonic measure ωin a domain Ω ⊂Rd. Let us recall some results in this direction. A famous result due to Makarov ([Mak85]) states that the harmonic measure in a simply connected domain of R2is always supported by a set of Hausdorff dimension 1 while every set with dimension strictly less than 1 is negligible with respect to the harmonic measure. A few years later, Jones and Wolff ([JW88]) extended this result and proved that in a general domain in R2, the harmonic measure is always supported by a set of dimension one. When Ω is the complementary of a self-similar Cantor set, Carleson proved that the dimension of the harmonic measure ωsatisfies dim∗(ω) = dim∗(ω)<dim(∂Ω). In that case, the harmonic measure can be seen as a Gibbs measure on a symbolic Cantor set and the properties of the harmonic measures are consequences of Ergodic theory (see also [MV86]). Such approach was also used in the more general situation of “conformal Cantor sets”, generalized snowflakes and Julia sets of hyperbolic polynomials (see the survey paper [Mak98] on this subject). In a nondynamical context, Batakis proved in [Bat96] the relation dim∗(ω)<dim(∂Ω) for a large class of domains Ω for which Ωcis a Cantor set. Let us finally recall Bourgain’s result in higher dimension: the harmonic measure is always supported by a set of dimension d−εwhere εonly depends on the dimension d(see [Bou87]). Explicit values of the dimension of measures can also often be obtained in dynamical contexts. This is for example the case for self-similar measures on self-similar Cantor sets. Let us briefly explain the calculus. Let On Dimension of Measures 247 S1,...,Skbe similarities in Rdwith ratio 0 < ri<1 and Ebe the unique nonempty compact set such that E=S i Si(E) (see [Hut81]). Suppose that the compact sets Si(E) are disjoint. Let p= (p1,...,pk) be a probability vector and mbe the unique probability measure such that (5) m= k X i=1 pim◦S−1 i. The measure mis nothing else but the image of a multinomial measure on the symbolic Cantor set {1,...,k}N∗through the homeomorphism i= (i1,...,in,...)∈ {1, . . . , k}N∗7−→ \ n Si1◦ · · · ◦ Sin(E). Let Ei1,...,in=Si1◦ · · · ◦ Sin(E). For every x∈Ethere exists a unique sequence i1(x),...,in(x),... such that x∈Ei1(x),...,in(x)for all n. Moreover, if fn i(x) is the frequency of the digit iin the sequence i1(x),...,in(x), we have log m(Ei1(x),...,in(x)) log diam(Ei1(x),...,in(x))=Pk i=1 fn i(x) log pi Pk i=1 fn i(x) log ri+1 nlog diam(E). Using the strong law of large numbers we get (6) lim n→+∞ log m(Ei1(x),...,in(x)) log diam(Ei1(x),...,in(x))=Pk i=1 pilog pi Pk i=1 pilog ri dm-almost surely. If we observe that Ei1(x),...,in(x)is in some sense similar to the ball of center xand radius diam(Ei1(x),...,in(x)), we get dim m(x) = dim m(x) = Pk i=1 pilog pi Pk i=1 pilog ri dm-almost surely and we can conclude that (7) dim∗(m) = dim∗(m) = Pk i=1 pilog pi Pk i=1 pilog ri . This formula is always true when the Open Set Condition is satisfied (see Part 1 for an elementary example). The calculus is much more complicated (and often impossible) in “overlapping” situations (see for example [LN98], [FL02], [Fen03], [Tes04], [Tes06a]). More generally, the thermodynamic formalism and the ergodic theory are in practice good tools to compute the dimension of measures. 248 Y. Heurteaux Let us for example mention the nice paper of L. S. Young in which a formula (involving the entropy and the Lyapunov exponents) is given for the upper dimension of invariant ergodic measures with respect to a C1+αdiffeomorphism of a compact surface ([You82]). Multifractal analysis is the natural way to obtain a more precise analysis of the measure m. The object is to compute the spectrum, defined as the following function: d(α) = dim x; dim m(x) = dim m(x) = α. In many situations, d(α) is nothing else but the Legendre transform τ∗(α) of the Lq-spectrum (8) τ(q) = lim sup n→+∞ τn(q) with τn(q) = 1 nlog ℓlog X I∈Fn m(I)q! where (Fn)n≥0are the natural partitions in dyadic (or ℓ-adic) cubes in Rd. When d(α) = τ∗(α), we say that the multifractal formalism is valid. A heuristic justification of the multifractal formalism runs as follows: First, the contribution to τn(q) of the set of points where the local exponents takes a value αis estimated. If the dimension of this set is d(α), then there are about ℓnd(α)cubes in Fnwhich cover this set and such a cube Isatisfies m(I)≈ℓ−αn. Therefore, the order of magnitude of the required contribution is ℓ−(αq−d(α))n. When ngoes to +∞, the maximum contribution is clearly obtained for the value of αthat minimizes the exponent αq −d(α); thus τ(q) = infα(αq −d(α)). If d(α) is a concave function, then this formula can be inverted and d(α) can be obtained from τ(q) by an inverse Legendre transform: (9) d(α) = inf q(αq +τ(q)). There are many papers who support formula (9). Frisch and Parisi ([FP85]) were the first to introduce the Legendre transform in multifractal analysis. Rigorous approaches are given by Brown, Michon Peyri`ere ([BMP92]) and Olsen ([Ols95]). They enlighten the link between formula (9) and the existence of auxiliary measures mqsatisfying (10) 1 Cm(I)q|I|τ(q)≤mq(I)≤C m(I)q|I|τ(q). In fact, it is shown in [Ben94], [BBH02] that the existence of a measure mqsatisfying (11) mq(I)≤C m(I)q|I|τ(q) On Dimension of Measures 249 is sufficient to obtain the nontrivial inequality d(α)≥inf q(αq +τ(q)). Now again, the dynamical context is a paradigm for multifractal analysis. In many situations, the existence of measures mqsatisfying (10) and the validity of (9) are proved. This is for example the case for quasiBernoulli measures ([BMP92], [Heu98], [Pey92]), self-similar measures ([CM92], [Fen03], [Fen05], [FO03], [HL01], [LN98], [LN00], [Rie95], [Tes06a], [Ye05]), measures on cookie-cutters ([Ran89]), graph-directed constructions ([EM92]), invariant measures of rational maps on the complex plane ([Lop89]). The context of self-affine measures is much more complicated ([Kin95], [Ols98]). The case of random self-similar measures was also studied ([Man74], [KP76], [Bar99], [Bar00a], [Bar00b]). Let us briefly explain the ideas that are used to validate the multifractal formalism in the context of self-similar measures on a self-similar Cantor set. The notations are the same as before (see (5) and the notations below). The partitions given by the compact sets Ei1,...,inare prefered to the (Fn)n≥0. In fact, it is easy to show that the measure m is doubling and that the sequence Ei1(x),...,in(x)of neighborhoods of x calculates the local exponents at point x. Let q∈Rand let τ=τ(q) be the unique real number such that (12) k X i=1 pq irτ i= 1. The function τ=τ(q) is similar to the Lq-spectrum defined by (8). The function τis convex and real analytic. Let mqbe the self-similar measure such that for all i, mq(Ei) = pq irτ i. The measure mqis such that for all i1,...,in, mq(Ei1,...,in)=(pi1···pin)q(ri1···rin)τ≈m(Ei1,...,in)qdiam (Ei1,...,in)τ, which is similar to (10). Let α=−τ′(q) = Pk i=1 pq irτ ilog pi Pk i=1 pq irτ ilog ri and Eα=(x∈E; lim n→+∞ log mEi1(x),...,in(x) log diam Ei1(x),...,in(x)=α). 250 Y. Heurteaux We observe that x∈Eαif and only if lim n→∞ log mEi1(x),...,in(x) log diam Ei1(x),...,in(x)=αq +τ(q) = Pk i=1 pq irτ ilog(pq irτ i) Pk i=1 pq irτ ilog ri . Applying (6) and (7) to the measure mq, we obtain dim(Eα) = dim(mq) = −qτ′(q) + τ(q) = inf q(αq +τ(q)) which is the desired formula. This example points out the importance of auxiliary measures in the multifractal analysis. In Part 5, we will apply the same technique to quasi-Bernoulli measures. The purpose of this survey paper is to revisit the notion of dimension of a measure in a very simple way. We do not refer to any dynamical context and we try to obtain estimates of the lower and the upper dimension which are always true. The probabilistic interpretation of the notion of dimension will be useful to achieve our purpose. As it is shown in Part 3, the lower and the upper dimension of a measure mare related to the asymptotic behaviour of a sequence of random variables. More precisely, if (Fn)n≥0are the natural partitions in dyadic (or ℓ-adic) cubes in Rdand if In(x) is the unique cube that contains x, we will see that the lower dimension (resp. upper dimension) of the measure mcoincides with the lower essential bound (resp. upper essential bound) of the random variable lim inf n→+∞Sn/n, where Sn n=X1+···+Xn nand Xn(x) = −logℓm(In(x)) m(In−1(x)). Similar interpretation of Dim∗(m) and Dim∗(m) in terms of the essential bounds of lim sup n→+∞ Sn/n is also possible. It is then not surprising that the lower and the upper dimension of the measure mare related to the log-Laplace transform of the sequence Sn: L(q) = lim sup n→+∞ 1 nlogℓEℓqSn, where the expectation is related to the probability m. An easy calculation gives L(1 −q) = lim sup n→+∞ 1 nlog ℓlog X I∈Fn m(I)q!:= τ(q) On Dimension of Measures 251 where we recognize the classical Lq-spectrum τused in multifractal analysis. The lower and the upper entropy of the measure mcan also be expressed in terms of the sequence of random variables Sn. More precisely, we have h∗(m) := lim inf n→+∞ −1 nX I∈Fn m(I) logℓm(I) = lim inf n→+∞ ESn n h∗(m) := lim sup n→+∞ −1 nX I∈Fn m(I) logℓm(I) = lim sup n→+∞ ESn n and these quantities are also related to the dimension of the measure m. All those estimates are gathered in Theorem 3.1 which states that (13) −τ′ +(1) ≤dim∗(m)≤h∗(m)≤h∗(m)≤Dim∗(m)≤ −τ′ −(1). A probabilistic interpretation of (13) is proposed in Theorem 3.2 and the equality cases are discussed in Part 4. Classical examples and concrete estimates are also developed to illustrate the purpose. In the last part (Part 5), we revisit the notion of quasi-Bernoulli measures in order to explain the importance of the estimates that are developed in the previous sections. Ergodicity properties are explained, the existence of the derivative function τ′is shown and an elementary proof of the validity of the multifractal formalism is given. Such a proof points out the important role played by the dimension of auxiliary measures in multifractal analysis. 1. A classical example: the Bernoulli product We begin this paper with the study of a classical example. It is a convenient way to introduce the notion of dimension of measures and to precise some notations. Moreover, generalizations of this example will be developed later (see Part 3.1). Let Fnbe the family of dyadic intervals of the nth generation on [0,1), 0< p < 1 and let mbe the Bernoulli product of parameter p. It is defined as follows. If ε1···εnare integers in {0,1}, and if Iε1···εn="n X i=1 εi 2i, n X i=1 εi 2i+1 2n!∈ Fn then m(Iε1···εn) = psn(1 −p)n−sn,where sn=ε1+···+εn. 258 Y. Heurteaux The random variables εiare independent and verify m({εn= 1}) = pnand m({εn= 0}) = 1 −pn. The random variables Xn, defined by (16) are independent and bounded in L2. The strong law of large numbers ensures that the sequence (19) Sn−E[Sn] n is almost surely converging to 0. We can easily conclude that for dm-almost every x∈[0,1), lim inf n→∞ log m(In(x)) log |In(x)|= lim inf n→∞ ESn n = lim inf n→∞ −1 n n X k=1 pklog2pk+ (1 −pk) log2(1 −pk). We write h∗(m) = lim infn→∞ ESn n. This quantity is called the lower entropy of the measure m(see Section 3.2). In this case, the measure mis always a unidimensional measure with dimension dim(m) = h∗(m). More precisely, we can deduce from (19) the existence of a subsequence nksuch that for almost every x∈[0,1), lim k→+∞ log m(Ink(x)) log |Ink(x)|=h∗(m). We will see in Section 4.3 that this kind of property characterizes measures for which the dimension can be calculated with an entropy formula. Of course, a similar result can be written with packing dimensions. The measure mis unidimensional and satisfies Dim(m) = lim sup n→∞ ESn n = lim sup n→∞ −1 n n X k=1 pklog2pk+ (1 −pk) log2(1 −pk) := h∗(m). Note that we may have dim(m)6= Dim(m). On Dimension of Measures 259 3.2. The function τ, probabilistic interpretation and links with entropy. Relations (17) and (18) do not help to find the dimensions of the measure m. From now on we try to obtain estimates of the quantities dim∗(m), dim∗(m), Dim∗(m), Dim∗(m) and describe some equality cases. Let us introduce the function τwhich is well known in multifractal analysis. It is defined as (20) τ(q) = lim sup n→+∞ τn(q) with τn(q) = 1 nlog ℓlog X I∈Fn m(I)q! where mis a probability measure on [0,1)d. The function τis finite on [0,+∞) and may be degenerated on the open interval (−∞,0). It is convex, non increasing on its definition domain. If we equip the set [0,1)d with the probability m, we can write: (21) τn(1−q)= 1 nlogℓEℓqSnand τ(1−q)=lim sup n→∞ 1 nlogℓEℓqSn. Taking the derivative, we get −τ′ n(1) = ESn n=−1 nX I∈Fn m(I) logℓm(I). This quantity is nothing else but the entropy of the probability mrelated to the partition Fn. It will be denoted by hn(m). In a general setting the sequence hn(m) does not necessarily converge. Nevertheless, one can always define the lower and the upper entropy with the formula (22) h∗(m) = lim inf n→∞ hn(m) and h∗(m) = lim sup n→∞ hn(m). If h∗(m) = h∗(m), the common value is denoted by h(m). It is the entropy of the measure m. Let us remark that convexity properties ensure that (23) −τ′ +(1) ≤h∗(m)≤h∗(m)≤ −τ′ −(1), where τ′ −et τ′ +are respectively the left and the right derivative of the convex function τ. 260 Y. Heurteaux Let us finish this section with the example described in Part 3.1. Easy calculations give τ(q) = lim sup n→+∞ 1 n n X k=1 log2(pq k+ (1 −pk)q) h∗(m) = lim inf n→∞ −1 n n X k=1 pklog2pk+ (1 −pk) log2(1 −pk) h∗(m) = lim sup n→∞ −1 n n X k=1 pklog2pk+ (1 −pk) log2(1 −pk). In particular, if mis a Bernoulli product with parameter p(that is, if pk=pfor all k), we get τ(q)=log2(pq+ (1 −p)q) and h(m)=−(plog2(p)+(1−p) log2(1−p)). 3.3. General estimates. There are deep links between the function τ, entropy and the dimension of the measure m. These can be resumed in the following theorem. Theorem 3.1 ([Heu98], [BH02]).Let mbe a probability measure on [0,1)d. We have (24) −τ′ +(1) ≤dim∗(m)≤h∗(m)≤h∗(m)≤Dim∗(m)≤ −τ′ −(1). Remarks. 1. In particular, (17) and (18) ensure that if dim∗(m) = Dim∗(m), then the entropy h(m) exists and lim n→∞ −logℓm(In(x)) n=h(m), dm-almost surely. We then obtain some kind of “Shannon-McMillan conclusion” in a non dynamical context. It is in particular the case if τ′(1) exists. 2. Conversely, if there exists a real number hsuch that lim n→∞ −logℓm(In(x)) n=halmost surely, we have dim∗(m) = Dim∗(m) and h∗(m) = h∗(m) = h. 3. In [Nga97], S.-M. Ngai proves inequalities like −τ′ +(1) ≤dim∗(m) and Dim∗(m)≤ −τ′ −(1). His purpose is then to consider the case where τ′(1) exists. Here we will first consider the non differentiable case (see Parts 3.4 and 4.2) and then find conditions that ensure that τ′(1) exists (see Part 5). On Dimension of Measures 261 Formulas (17) and (18) give links between the dimension of the measure mand the asymptotic behavior of the sequence Sn/n. They allow us to propose a very simple proof of Theorem 3.1. This is not the way used in [Heu98] but we can isolate the following result which immediately gives Theorem 3.1. Theorem 3.2. Let (Sn)n≥0be a sequence of random variables on a probability space (Ω,A,P). Suppose that the function L(q) = lim sup n→∞ 1 nlogℓEℓqSn is finite on a neighborhood Vof 0. Then we have: L′ −(0) ≤ess inf lim inf n→∞ Sn nand ess sup lim sup n→∞ Sn n≤L′ +(0). Moreover, the sequence Sn nis dominated in L1(P)and ess inf lim inf n→+∞ Sn n≤lim inf n→+∞ ESn n ≤lim sup n→+∞ ESn n≤ess sup lim sup n→+∞ Sn n. Proof of Theorem 3.2: Let α > L′ +(0) and q > 0. Using Cramer-Chernov’s idea, we have PSn n≥α≤1 ℓqnα EℓqSn. Taking the logarithm and the lim sup, we get lim sup n→∞ 1 nlogℓPSn n≥α≤L(q)−qα and we can conclude that lim sup n→∞ 1 nlogℓPSn n≥α≤ − sup q>0, q∈V (qα −L(q)) = −L∗(α)<0, where L∗is the Legendre transform of L. If 0 < ε < L∗(α) and if nis sufficiently large, we obtain PSn n≥α≤e−n(L∗(α)−ε). Then, Borel-Cantelli’s lemma gives Plim sup n→∞ Sn n≥α= 0, 262 Y. Heurteaux which clearly implies that lim supn→+∞Sn n≤αalmost surely. The inequality ess sup lim sup n→∞ Sn n≤L′ +(0) follows. With a similar argument, we can also prove the other inequality L′ −(0) ≤ess inf lim inf n→∞ Sn n. In order to obtain the second point of the theorem, we first observe that the sequence Sn nis dominated in L1(P). Indeed, let X= supnSn n. We have: P(X > t)≤X n≥1 P Sn n > t=X n≥1 PSn n> t+PSn n<−t. On the other hand, if q > 0 is such that L(q)<+∞and if ε > 0, the preceding calculus allows us to find an integer n0such that for every n≥ n0, 1 nlogℓPSn n> t≤L(q) + ε−qt. If tis large enough, we get X n≥n0 PSn n> t≤X n≥n0 ℓn(L(q)+ε−qt)≤ℓL(q)+ε−qt 1−ℓL(q)+ε−qt which proves that the function t7→ X n≥1 PSn n> t is integrable with respect to the Lebesgue’s measure. A similar result is true for the function t7→ Pn≥1PSn n<−t. Finally, E[X] = Z+∞ 0 P(X > t)dt < +∞. Having just proved that the sequence Sn nis dominated in L1(P) by the random variable X, Fatou’s lemma applied to the positive sequence X+ On Dimension of Measures 263 Sn ngives E[X] + ess inf lim inf n→+∞ Sn n=EX+ ess inf lim inf n→+∞ Sn n ≤EX+lim inf n→+∞ Sn n ≤lim inf n→+∞ EX+Sn n =E[X] + lim inf n→+∞ ESn n, and the first inequality follows. In order to prove the second inequality, it suffices to apply Fatou’s lemma to the positive sequence X−Sn n. 3.4. How to use Theorem 3.1. In general it is awkward or even impossible to obtain exact values for the function τand the numbers τ′ −(1) and τ′ +(1). Nevertheless, if we can estimate in a neighborhood of 1 the function τby a function χ satisfying χ(1) = 0, we obtain dim∗(m)≥ −χ′ +(1) and Dim∗(m)≤ −χ′ −(1). In particular, this remark can be applied to χ= logℓ(β) where β(q) = lim sup n→+∞ βn(q) and βn(q) = sup I∈Fn X J⊂I, J∈Fn+1 m(J) m(I)q . This is a consequence of the inequalities τ(q)≤lim sup n→+∞ log βn−1(q) + ···+ log β0(q) nlog ℓ ≤lim sup n→+∞ log βn(q) log ℓ=log β(q) log ℓ. Finally, using β(1) = 1, we get the following corollary: Corollary 3.3 ([Heu95], [Heu98]).Let mbe a probability measure on [0,1)dand βdefined as above. We have dim∗(m)≥ −β′ +(1) ln(ℓ)and Dim∗(m)≤ −β′ −(1) ln(ℓ). 264 Y. Heurteaux 3.5. Contrasts and dimension’s estimates. The function βngives estimates of the contrasts between the mass of a cube Iand the mass of its sons. In numerous situations, those contrasts can be estimated and we can then deduce estimates of the dimension of the measure. In particular, this is what is done by Bourgain in [Bou87] and Batakis in [Bat96] when they give estimates of the dimension of the harmonic measure. Some elementary situations, which are particular cases of Propositions 3.4 and 3.5 are also proposed in [Heu95]. Let us describe a general way to obtain concrete estimates. Suppose that every cube I∈SnFnhas a positive mass. Let k∈ {1,...,ℓd−1} and if I∈ Fn,n≥1, let δk(I) = max m(I1∪ · · · ∪ Ik) m(I), I1,...,Iksons of I. We first remark that if J1,...,Jℓdare the sons of Iand satisfy m(J1)≥ · · · ≥ m(Jℓd), we have δk(I) = m(J1∪ · · · ∪ Jk) m(I)and ∀j > k, km(Jj)≤m(J1∪ · · · ∪ Jk). It follows that 1 = δk(I) + X j>k m(Ji) m(I)≤δk(I) + (ℓd−k) kδk(I) and we can claim that (25) k ℓd≤δk(I)≤1. If δk(I)≈k ℓd, the measure mis quite homogenous in the cube I. If it is true in every cube, we can hope that the dimension of mis big. On the other hand, if for every cube I,δk(I)≈1, a small part of Icontains a large part of the mass and we can hope that the dimension of mis small. These remarks can be made precise in the following propositions. Proposition 3.4. Let mbe a probability measure on [0,1)d,1≤k < ℓd and kℓ−d< δ < 1such that for every I∈SnFn,δk(I)≥δ. Then, the measure msatisfies Dim∗(m)≤ −δlogℓδ k−(1 −δ) logℓ1−δ ℓd−k. On Dimension of Measures 265 Proposition 3.5. Let mbe a probability measure on [0,1)d,1≤k < ℓd and kℓ−d< δ < 1such that for every I∈SnFn,δk(I)≤δ. Let p=δ−1. Then, the measure msatisfies dim∗(m)≥ −pδ logℓ(δ)−(1 −pδ) logℓ(1 −pδ). Proposition 3.5 is in fact an elementary consequence of the more general following result. Proposition 3.6. Let mbe a probability measure on [0,1)dand 0<δ≤1. Let p=δ−1and suppose that for every cube I∈SnFn, we can find a partition A1,...,Ajof the set of sons of Isuch that ∀i∈ {1,...,j},mSJ∈AiJ m(I)≤δ. Then dim∗(m)≥ −pδ logℓ(δ)−(1 −pδ) logℓ(1 −pδ). Remark 6.If δ > 1/2, then p=1. This is in particular the case when ℓ=2 and d= 1. Remark 7.When k= 1 and ℓ= 2, similar estimations are also obtained by Gonz´alez Llorente and Nicolau in [LN04]. Logarithm corrections are also proposed. Proof of Proposition 3.4: This proposition can be found in [Heu98]. Let us sketch the proof in order to be self contained. Let I∈ Fnand I1,...,Ikthe sons of Isuch that δk(I) = m(I1∪···∪Ik) m(I). Denote S= {I1,...,Ik}. If q < 1, H¨older’s inequality gives X J⊂I, J∈Fn+1 m(J) m(I)q =X J∈Sm(J) m(I)q +X J6∈Sm(J) m(I)q ≤k1−q(δk(I))q+ (ℓd−k)1−q(1 −δk(I))q. Let us observe that the function t7→ k1−qtq+ (ℓd−k)1−q(1 −t)qis decreasing on the interval [kℓ−d,1]. Under the hypothesis of Proposition 3.4, we obtain ∀q∈]0,1[, βn(q)≤k1−q(δ)q+ (ℓd−k)1−q(1 −δ)q, and the conclusion follows from Corollary 3.3. Proof of Proposition 3.6: We begin with the following lemma. 266 Y. Heurteaux Lemma 3.7. Let q > 1,j≥2and 1 j< δ ≤1. Denote by M(δ, j) the maximum of the function F(a1,...,aj) = aq 1+···+aq junder the constraints a1+···+aj= 1 and 0≤ai≤δ,∀i. Then M(δ, j) = pδq+ (1 −pδ)q where p=δ−1. Proof: The function Fbeing symmetric, we can add the constraint a1≥ · · · ≥ aj. Observe that we have j≥p+ 1. If 0 < a2≤a1< δ, the function ε > 07→ (a1+ε)q+ (a2−ε)qis increasing, so that the maximum is obtained when a1=δ. We then prove the lemma by recurrence on the integer p. Suppose first that p= 1, that is 1 2< δ ≤1. We have F(δ, a2,...,aj)≤δq+ (a2+···+aj)q=δq+ (1 −δ)q. Moreover, under the hypothesis p= 1, we have 0 ≤1−δ < δ,F(δ, 1− δ, 0,...,0) = δq+(1−δ)qand we can conclude that M(δ, j) = δq+(1−δ)q. Suppose now that the conclusion of the lemma is satisfied for every value of δ−1between 1 and p−1 and let δsuch that δ−1=p. The real number δsatisfies the inequalities 1 p+1 < δ ≤1 pand we observe that F(δ, a2,...,aj) = δq+ (1 −δ)q a2 1−δq +···+aj 1−δq. The real numbers ai 1−δsatisfy the constraints 0≤ai 1−δ≤δ 1−δ. Moreover, 1−δ δ=p−1 and 1 j−1<δ 1−δ. We can then use the recurrence hypothesis and obtain F(δ, a2,...,aj)≤δq+Mδ 1−δ, j −1 =δq+(1 −δ)q(p−1)δ 1−δq +1−(p−1) δ 1−δq =pδq+ (1 −pδ)q. It follows that M(δ, j)≤pδq+ (1 −pδ)q. In fact, the last inequality is an equality if we remark that 1 −pδ ≤δand F(δ, . . ., δ, (1 −pδ),0,...,0) = pδq+ (1 −pδ)q. On Dimension of Measures 267 We can now finish the proof of Proposition 3.6. We want to estimate the function βof Part 3.4. Let I∈ Fn. If q > 1, Lemma 3.7 ensures that X J⊂I, J∈Fn+1 m(J) m(I)q = j X i=1 X J∈Aim(J) m(I)q ≤ j X i=1 mSJ∈AiJ m(I)!q ≤pδq+ (1 −pδ)q. We can deduce that β(q)≤pδq+ (1 −pδ)qif q > 1 and conclude that dim∗(m)≥ −β′ +(1) log ℓ≥ −pδ logℓ(δ)−(1 −pδ) logℓ(1 −pδ). 4. Situations where it is possible to obtain an exact formula for the dimension 4.1. Equalities −τ′ −(1) = Dim∗(m) and −τ′ +(1) = dim∗(m) are often false. In general −τ′ +(1) 6= dim∗(m) and −τ′ −(1) 6= Dim∗(m). For example, Olsen in [Ols00] gives an example of a discrete measure such that −τ′ −(1) = 1 and −τ′ +(1) = 0. We give here a more convincing example. Proposition 4.1. Let µbe a continuous measure with support [0,1]. We can construct a measure mwhich is equivalent to µand for which the function τsatisfies τ(q) = sup(1 −q, 0) if q > 0. In particular, the measures µand mhave the same dimensions but the function τassociated to mis degenerated. Applying this proposition to a Bernoulli product for which the parameter psatisfies −(plog2(p) + (1 −p) log2(1 −p)) = h, we obtain the following corollary. 274 Y. Heurteaux Recalling that Zn≥0, we get δ+η2≥E[Zn1] ≥ZA\B Zn1dP+ZB Zn1dP ≥(δ−η2)(P[A]−P[B]) + (δ+η)P[B]. Moreover, P[A]≥1−η2, so that P[B]≤2η2+δη2 η+η2≤(2 + δ)η. In order to prove Theorem 4.7, we use Lemma 4.8 with η= 2−kand then construct a subsequence nksuch that ∀k, PZnk> δ + 2−k≤(2 + δ)2−k. Using Borel-Cantelli’s lemma, we deduce that lim sup k→+∞ Znk≤δ dP-almost surely. Moreover δ=S∗≤lim inf k→+∞ZnkdP-almost surely and we can conclude that the subsequence Znkis almost surely converging to δ. The proof is finished if we observe that under hypothesis (ii), Z∗= ess inf(Z∗) = δ dP-almost surely. 4.4. Entropy is a bad notion of dimension. Entropy can not allow us to classify measures. For example, there exist equivalent probability measures with different entropies. Let us precise this phenomenon in the following example. Proposition 4.9. Let m0and m1be two probability measures on [0,1)d such that the entropies h(m0)and h(m1)exist and are different. If 0< α < 1, let mα=αm1+ (1 −α)m0. Then, h(mα) = αh(m1) + (1 −α)h(m0). In particular, the family (mα)0<α<1is constituted of equivalent measures for which entropy varies in a non trivial interval. On Dimension of Measures 275 Proof: The notations are the same as in Part 3.2. We remark that the function x7→ −xlogℓ(x) is concave. It follows that hn(mα)≥αhn(m1) + (1 −α)hn(m0), and (33) h∗(mα)≥αh(m1) + (1 −α)h(m0). On the other hand, if q < 1 and if xand yare two positive numbers, it is well known that (αx + (1 −α)y)q≤αqxq+ (1 −α)qyq. We can deduce that X I∈Fn m(I)q≤αqX I∈Fn m1(I)q+ (1 −α)qX I∈Fn m0(I)q. These two quantities are equal to 1 if q= 1. We can then take the derivative at q= 1 and obtain hn(mα)≤αhn(m1)−αlogℓα n+ (1 −α)hn(m0)−(1 −α) logℓ(1 −α) n. Finally, (34) h∗(mα)≤αh(m1) + (1 −α)h(m0). Inequalities (33) and (34) give the conclusion of Proposition 4.9. 5. Quasi-Bernoulli measures In this section, we suppose for simplicity that d= 1. The notations are the same as in Section 4.2. We say that the probability measure m is a quasi-Bernoulli measure if we can find C≥1 such that (35) ∀I, J ∈[ n Fn,1 Cm(I)m(J)≤m(IJ)≤C m(I)m(J). Quasi-Bernoulli property does appear in many situations. In particular, this is the case for the harmonic measure in regular Cantor sets ([Car85], [MV86]) and for the caloric measure in domains delimited by Weierstrass type graphs ([BH00]). Let us introduce the natural applications between [0,1) and the Cantor set C={0,...,ℓ−1}N∗: J: [0,1) −→ C and S:C −→ [0,1]. They are defined by: J(x) = (εi)i≥1if {x}=\ n Iε1···εnand S((εi)i≥1) = \ n ¯ Iε1···εn. 276 Y. Heurteaux The application Jis a bijection between [0,1) and the complement of a countable subset of C. Observing that a quasi Bernoulli measure does not contain any Dirac mass, we can carry the measure mthrough the application Jand work on the Cantor set C. We always denote by m this new measure and every property that is proved for this new measure can be pulled back. Let Mbe the set of words written with the alphabet {0,...,ℓ−1}. There is a link between the words of Mand the cylinders in the Cantor set C, so that Property (35) can be rewritten (36) ∀a, b ∈ M,1 Cm(a)m(b)≤m(ab)≤C m(a)m(b). (ab is the concatenation of the words aand b.) We say that the measure mis a quasi Bernoulli measure on the Cantor set C. Let Mnbe the set of words of length n, and if x=x1x2· · · ∈ C, let In(x) = x1···xnbe the unique cylinder Mnthat contains x. In this new context, it is always possible to define τnand τ. Submultiplicative properties like in Part 4.2 ensure that the sequence τn(q) is convergent when mis a quasi-Bernoulli measure. We then have (37) τ(q) = lim n→+∞τn(q) with τn(q) = 1 nlog ℓlog X a∈Mn m(a)q!, and the following inequalities are true (38) C−|q|ℓnτ(q)≤X a∈Mn m(a)q=ℓnτn(q)≤C|q|ℓnτ(q). Let us finally remark that we can suppose that for every a∈ M, m(a)>0. Indeed, if it is not the case, quasi-Bernoulli property ensures that there exists a cylinder a∈ M1such that m(a) = 0. Finally, several letters are not useful in the alphabet and one can work in a smaller Cantor set. 5.1. 0-1 law and mixing properties. The interest in working on the Cantor set Cis the dynamical context related to the shift (39) σ: (εn)n≥1∈ C 7−→ (εn)n≥2∈ C. In particular, if a∈ Mn, then ab =a∩σ−n(b). We can isolate the following properties that precise some previous remarks due to Carleson and Makarov-Volberg ([Car85], [MV86]). On Dimension of Measures 277 Proposition 5.1. Let mbe a quasi-Bernoulli measure on the Cantor set C. Let B0be the σ-field of Borel sets , Bn=σ−n(B0)and B∞= TnBn. (i) For every E∈B∞,m(E) = 0 or m(E) = 1. (0-1law). (ii) Moreover, if mis σ-invariant, the strong mixing property is true. That is ∀A, B ∈B0,lim n→∞ mA∩σ−n(B)=m(A)m(B). Remark. In particular, every σ-invariant quasi-Bernoulli measure is ergodic. Proof: Let E∈B∞be such that m(E)>0. For every n∈Nwe can find a Borel set Fsuch that E=σ−n(F). We can also find a cylinder a0∈ Mnsuch that m(a0∩E) m(a0)≥1 2m(E). Quasi-Bernoulli property ensures that ∀a∈ Mn,∀b∈ M,m(a∩σ−n(b)) m(a)≥1 C2 m(a0∩σ−n(b)) m(a0). Observing that an open set is the union of a countable family of disjoint cylinders, the previous inequality is also true if bis an open set. Finally, using the regularity properties of the measure m, it is true for every Borel set b. Replacing bby F, we obtain ∀a∈ Mn,m(a∩E) m(a)≥1 C2 m(a0∩E) m(a0)≥1 2C2m(E). A similar argument proves that the inequality m(a∩E)≥(2C2)−1m(E)m(a) is also true for every Borel set a. In particular, m((C \ E)∩E)≥(2C2)−1m(E)m(C \ E), which says that m(C \ E) = 0. That is what we wanted to prove. The proof of (ii) is then classical. Let Zn=E[11A|Bn]. It is a martingale with respect to de decreasing sequence of σ-fields Bn. It is converging in the L2sense (and also almost-surely) to Z∞=E[11A|B∞]. But B∞is the trivial σ-field. Then Z∞is a constant random variable. Moreover, E[Zn] = m(A). Taking the limit, we get Z∞=E[Z∞] = m(A)dm-almost-surely. 278 Y. Heurteaux Finally, m(A∩σ−n(B)) −m(A)m(B)=E11A11σ−n(B)−Em(A)11σ−n(B) =E(Zn−Z∞)11σ−n(B) ≤Eh|Zn−Z∞|2i1/2, and the strong mixing property is proved. Let us now introduce the following definition. Definition 5.2. Let m1and m2be two probability measures on C. We say that m1and m2are strongly equivalent if we can find c > 0 such that: 1 cm1≤m2≤c m1. We then have the following corollary. Corollary 5.3. Let mbe a quasi-Bernoulli measure on C. There exists a unique probability measure, which is quasi-Bernoulli, σ-invariant and strongly equivalent to m. Moreover, it is obtained as the weak limit of the sequence mndefined by mn(E) = 1 n n X k=1 mσ−k(E). Proof: Observe that every probability measure which is strongly equivalent to a quasi-Bernoulli measure is also a quasi-Bernoulli measure. Moreover, it is well known that two equivalent ergodic probabilities are equal. These two facts prove the uniqueness. In order to prove the existence, we first compare the measures mn and m. If a∈ M, we have: m(σ−k(a))=m [ b∈Mk ba!=X b∈Mk m(ba)≤CX b∈Mk m(b)m(a)=Cm(a). It follows that mn≤Cm with a constant Cthat does not depend on n. The inequality mn≥1 Cmis also true. We can then deduce that the measures mnare quasi-Bernoulli with a constant that does not depend on n. It follows that every weak limit of a subsequence mnkis quasiBernoulli and strongly equivalent to m. Let us finally consider an adherent value µof the sequence mnand a subsequence mnkwhich is weakly convergent to µ. If fis a continuous On Dimension of Measures 279 function on C, then Zf◦σ(x)dmnk(x) = 1 nk nk X j=1 Zf◦σj+1(x)dm(x) =Zf(x)dmnk(x)+ 1 nkZf◦σnk+1(x)dm(x)−Zf◦σ(x)dm(x). Taking the limit, we obtain Rf◦σ(x)dµ(x) = Rf(x)dµ(x), which says that µis σ-invariant. Finally, using the uniqueness, there is only one adherent value for the sequence mn. Then, the sequence mnis converging. 5.2. Showing that τis differentiable at point 1. Corollary 5.3, Theorem 4.3 and the Shannon-McMillan’s theorem allow us to prove that τ′(1) exists. This was done in [Heu98]. Theorem 5.4. Let mbe a quasi-Bernoulli measure on C. Quantities τ′(1) and h(m)exist and we have lim n→∞ −logℓm(In(x)) n=−τ′(1) = h(m)dm-almost surely. Remark. If the Cantor set Cis equipped with the natural ultra metric which gives the diameter ℓ−nto each cylinder in Mn, then −logℓm(In(x)) n is nothing else but the quotient of the logarithm of the mass of In(x) and the logarithm of its diameter. So, the measure mis unidimensional with dimension dim(m) = −τ′(1) = h(m). Let us now introduce the sets (40) Eα=x∈ C; lim n→∞ −logℓm(In(x)) n=α. Using Billingsley’s theorem (see [Fal90]), Theorem 5.4 shows that (41) dim(E−τ′(1)) = dim(m) = −τ′(1). This is the first step in the multifractal analysis of the measure m. Proof of Theorem 5.4: Let µbe the unique quasi-Bernoulli probability which is strongly equivalent to mand σ-invariant. The measures m and µhave the same function τand the same dimensions. Moreover, results of Part 4.2 can be applied to the measures mand µ. It follows that dim∗(m) = dim∗(µ) = −τ′ +(1) and Dim∗(m) = Dim∗(µ) = −τ′ −(1). 280 Y. Heurteaux Let us apply Shannon-McMillan’s theorem (see [Zin97]) to the measure µ. It says that the entropy h(µ) = lim n→+∞ −1 nX a∈Mn µ(a) logℓ(µ(a)) exists and that for dµ almost every x=x1x2· · · ∈ C, (42) −logℓµ(In(x)) n=−logℓµ(x1···xn) n−−−−−→ n→+∞h(µ). So, the measure µis unidimensional. Measures mand µbeing strongly equivalent, one can replace µby min (42). Finally, we have dim∗(m) = −τ′ +(1) = h(m) = −τ′ −(1) = Dim∗(m), which proves that τ′(1) exists. Let us finally remark that Theorem 5.4 and Corollary 3.3 allow us to deduce the following corollary. Corollary 5.5. Let mbe a quasi-Bernoulli probability on C. Let m0be the homogenous probability on Cwhich gives the mass ℓ−nto each cylinder in Mn. We have: dim(m) = 1 ⇐⇒ τ′(1) = −1⇐⇒ mis strongly equivalent to m0. Proof: Suppose that mis not strongly equivalent to m0. We can for example suppose that the inequality m0≤cm is never satisfied. We can then find an integer n0and a cylinder a0∈ Mn0such that m(a0)< 1 ℓC m0(a0) where Cis the constant which appears in the quasi-Bernoulli property. If a∈ M, we have m(aa0) m(a)≤1 ℓm0(a0) = ℓ−(n0+1). If 0 < q < 1, then X b∈Mn0 m(ab)q≤m(aa0)q+ (ℓn0−1) m(a)−m(aa0) ℓn0−1q ≤ ℓ−(n0+1)q+ (ℓn0−1) 1−ℓ−(n0+1) ℓn0−1q!m(a)q := γ(q)m(a)q. On Dimension of Measures 281 We can then sum this inequality on every cylinder of the same generation and then iterate the process. We get X a∈Mpn0 m(a)q≤(γ(q))p,∀p≥0, which gives τ(q)≤1 n0 logℓγ(q). Finally we have dim(m) = −τ′(1) ≤−γ′(1) n0log ℓ<1. 5.3. Multifractal analysis of quasi-Bernoulli measures. In [BMP92], Brown, Michon and Peyri`ere proved that the multifractal formalism is valid for quasi-Bernoulli measures at every point α which can be written α=−τ′(q). This result was one of the first rigorous results on multifractal analysis of measures. Unfortunately, they could not prove that the function τis of class C1. This has been done a few years later in [Heu98] and we can resume these two results in the following theorem. Theorem 5.6 ([BMP92], [Heu98]).Let mbe a quasi-Bernoulli measure on C. The function τis of class C1. Moreover, for every −τ′(+∞)< α < −τ′(−∞), dim(Eα) = τ∗(α) where the level set Eαis defined like in formula (40) and τ∗(α) = infq(αq +τ(q)) is the Legrendre transform of the function τ. Remark. In [Tes06a], Testud introduces a weaker notion which is called weak quasi-Bernoulli property. In this more general context, he proves that the function τis differentiable on [0,+∞) and satisfies dim(Eα) = τ∗(α) for every −τ′(+∞)< α < −τ′ +(0). Moreover, he also proves in [Tes06b] that the function τis not necessary differentiable on (−∞,0]. His results can be applied to a large class of self-similar measures with overlaps. 5.4. An easy proof of Theorem 5.6. We can give a proof of Theorem 5.6 which is much simpler than the original one and which points out the important role of auxiliary measures in multifractal analysis of measures. This approach is quite different to the one used in [BMP92] and [Heu98]. It was already present in 282 Y. Heurteaux my “m´emoire d’habilitation” [Heu99] but never published. It makes use of the relation between the real number τ′(1) (when it exists) and the asymptotic behavior of m(In(x)) (see Theorem 3.1 and the associated remarks). We begin with the construction of auxiliary measures mq,q∈R(so called Gibbs measures) which satisfy mq(a)≈m(a)q|a|τ(q)for every a∈ M(here |a|=ℓ−nif a∈ Mn). Lemma 5.7. Let q∈R. There exists a probability measure mqand a constant c≥1such that ∀a∈ M,1 cm(a)q|a|τ(q)≤mq(a)≤c m(a)q|a|τ(q). The measure mqis called the Gibbs measure at state q. Proof: In [Mic83], Michon proposed a construction of such measures. Let us present a simpler proof. Let us introduce some notation. If F1and F2are two functions which depend on qand on cylinders in M=SnMn, we will write F1≈F2if there exists a constant C > 0 which eventually depends on qbut which does not depend on the cylinders such that 1 CF1≤F2≤CF1. Let us first observe that ℓ(n+p)τn+p(q)=X a∈Mn, b∈Mp m(ab)q≈X a∈MnX b∈Mp m(a)qm(b)q=ℓnτn(q)ℓpτp(q). Let µnbe the unique measure such that µn(a) = m(a)q|a|τn(q)=m(a)qℓ−nτn(q)if a∈ Mnand which is homogenous on the cylinders of Mn. The measure µnis a probability measure. If a∈ Mnand if p≥1, we have µn+p(a) = X b∈Mp µn+p(ab) =X b∈Mp m(ab)qℓ−(n+p)τn+p(q) ≈m(a)qℓ−nτn(q)X b∈Mp m(b)qℓ−pτp(q) =m(a)qℓ−nτn(q). Moreover, we saw in (38) that ℓnτn(q)≈ℓnτ(q). Finally, ∀a∈ Mn,∀k > n, µk(a)≈m(a)qℓ−nτ(q)=m(a)q|a|τ(q). On Dimension of Measures 283 Let mqbe an adherent value of the sequence (µk)k≥1. The function 11a being continuous on the Cantor set C, we can take the limit and obtain (43) ∀a∈ M,1 cm(a)q|a|τ(q)≤mq(a)≤c m(a)q|a|τ(q), which finishes the proof of Lemma 5.7. We can now prove Theorem 5.6. An elementary computation shows that the function τassociated with the measure mq(which is denoted by τq) satisfies: τq(t) = τ(qt)−tτ(q). Moreover, mq(ab)≈m(ab)q|ab|τ(q)≈[m(a)m(b)]q(|a||b|)τ(q)≈mq(a)mq(b), which says that mqis a quasi-Bernoulli measure. The existence of τ′ q(1) proves the existence of τ′(q) and the relation −τ′ q(1) = −qτ′(q) + τ(q) = τ∗(−τ′(q)). Let α=−τ′(q). Inequality (43) ensures that Eα=x∈ C; lim n→∞ −logℓmq(In(x)) n=−τ′ q(1). Finally, Relation (41) written for the measure mqgives dim(Eα) = dim(mq) = −τ′ q(1) = τ∗(α). Of course, we need another argument to prove the existence of τ′(0). Taking the logarithm in (38), we have |τn(q)−τ(q)| ≤ |q|logℓC n. In particular, τn(0) = τ(0) and we deduce that  τn(q)−τn(0) q−τ(q)−τ(0) q ≤logℓC n. If q→0+and q→0−, we get     τ′ n(0) −τ′ +(0)≤logℓC n τ′ n(0) −τ′ −(0)≤logℓC n and we can conclude that τ′ +(0) = τ′ −(0). 290 Y. Heurteaux Laboratoire de Math´ematiques, UMR 6620 Universit´e Blaise Pascal F-63177 Aubi`ere France E-mail address:[email protected] Primera versi´o rebuda el 18 d’octubre de 2006, darrera versi´o rebuda el 16 de febrer de 2007.