Multifractal analysis of random weak Gibbs measures

We describe the multifractal nature of random weak Gibbs measures on some class of attractors associated with $C^1$ random dynamics semi-conjugate to a random subshift of finite type. This includes the validity of the multifractal formalism, the calculation of Hausdorff and packing dimensions of the so-called level sets of divergent points, and a $0$-$\infty$ law for the Hausdorff and packing measures of the level sets of the local dimension.


Introduction
Weak Gibbs measures are conformal probability measures obtained as eigenvectors of Ruelle-Perron-Frobenius operators associated with continuous potentials on topological dynamical systems. When the system (X, f ) has nice enough geometric properties, for instance in the case of a conformal repeller, these measures provide natural, and now standard examples of measures obeying the multifractal formalism: their Hausdorff spectrum and L q -spectrum form a Legendre pair.
Specifically, for such a measure µ on (X, f ), the (lower) L q -spectrum τ µ : R → R∪{−∞} is defined by (1) τ µ (q) = lim inf r→0 log sup{ i (µ(B i )) q } log(r) , where the supremum is taken over all families of disjoint closed balls B i of radius r with centers in supp(µ); the Hausdorff spectrum of µ is defined by where dim H stands for the Hausdorff dimension, E(µ, d) = {x ∈ supp(µ) : dim loc (µ, x) = d} , with dim loc (µ, x) = lim r→0 + log(µ(B(x, r))) log(r) , and we have the duality relation a negative dimension meaning that the set is empty. In fact, due to the super and submultiplicativity properties associated with µ, the same equality holds if we replace the limit by a lim inf or a lim sup in the definition of the local dimension. The rigorous study of these measures started with the Gibbs measures case, which corresponds to Hölder continuous potentials, or continuous potentials possessing the socalled bounded distorsions property, and in particular on the so-called "cookie-cutter" Cantor sets associated with a C 1+α expanding map f on the line [5,43] (see [41] for an extended discussion of dimension theory and multifractal analysis for hyperbolic conformal dynamical systems). This followed seminal works by physicists of turbulence and statistical mechanics pointing the accuracy of multifractals to statistically and geometrically describe the local behavior of functions and measures [17,19]. In the case of Gibbs measures, the L q -spectrum of the Gibbs measure is differentiable, and analytic if the potential φ is Hölder continuous; it is the unique solution t of the equation P (qφ + t log Df ) = 0, where P (·) stands for the topological pressure. The general case of continuous potentials was solved later in [36,10,20,13], with the same formula for the L q -spectrum. These progress then led to the multifractal analysis of Bernoulli convolutions associated with Pisot numbers [14,12]. Thermodynamic formalism and large deviations are central tool in these studies.
In the context of random dynamical systems, the multifractal analysis of random Gibbs measures (to be defined below) associated with random Hölder continuous potentials on attractors of random C 1+α expanding (or expanding in the mean) random conformal dynamics encoded by random subshifs of finite type has been studied in [23], [15] and [35]. These works, as well as the dimension theory of attractors of random dynamics [4,23,24,35], are based on the thermodynamic formalism for random transforms [22,3,4,18,21,6,27,7,35,8]. The multifractal analysis of random weak Gibbs measures is also implicitly considered in [15] (which deals with the multifractal analysis of Birkhoff averages), but the fibers are deterministic, and the techniques developed there seems difficult to adapt in a simple way in the case of random subshifts.
In this paper we consider, on a base probability space (Ω, F, P, σ), random weak Gibbs measures on some class of attractors included in [0, 1] and associated with C 1 random dynamics semi-conjugate (up to countably many points), or conjugate, to a random subshift of finite type. We provide a study of the multifractal nature of these measures, including the validity of the multifractal formalism, the calculation of Hausdorff and packing dimensions of the so-called level sets of divergent points, and a 0-∞ law for the Hausdorff and packing measures of the level sets of the local dimension. Compared to the above mentioned works, apart the source of new difficulties coming from the relaxation of the regularity properties of the potentials, our assumptions provide a more general process of construction of the random Cantor set in terms of the distribution of the random family of intervals used to refine the construction at a given step: it can contain contiguous intervals (i.e. without gap in between, and even no gap) with positive probability; thus, for instance, it covers the natural families of Cantor sets one can obtain by picking at random a fiber in a Bedford-McMullen carpet. Extensions of our results to the higher dimensional case will be discussed in Remark 1. We focus on the one dimensional case because our model will be used in a companion paper to study the multifractal nature of discrete measures obtained as "inverse" of the random weak Gibbs measures considered here.
Section 2 developes background about random dynamical systems and thermodynamic formalism, and presents our main results, namely theorems 2.6 and 2.7, as well as concrete examples of random attractors. Section 3 provides the basic properties that will used in the proof of this theorem in Section 4.

Setting and main result
We first need to expose basic facts from random dynamical systems and thermodynamic formalism.
For any s ∈ Σ ω,1 , n ≥ M (ω) and s ∈ Σ σ n ω,1 , there is at least one word v(s, s ) ∈ Σ σω,n−2 such that sv(s, s )s ∈ Σ ω,n . For each such connection, we fix one such v(s, s ) and denote the word sv(s, s )s by s * s .
where the supremum is taken over all finite or countable measurable partitions Q = {Q i } of Σ Ω with finite conditional entropy, that is h ρ (F, Q) < +∞. In our setting, we have h ρ (F ) ≤ log(l) dP. The number h ρ (F ), also denoted h(ρ|P) in the literature, is the relativized entropy of F given ρ. It is also called the fiber entropy of the bundle random dynamics F .

Multifractal analysis of the random weak Gibbs measures.
Our results statement require some additional definitions related to multifractal formalism.
Let µ be a compactly supported positive and finite Borel measure on R n .
and τ µ (q) = lim sup where the supremum is taken over all families of disjoint closed balls B i of radius r with centers in supp(µ).
By construction, the function τ µ is non decreasing and concave over its domain, which equals R of R + (see [29,2]).
Definition 2.2. The lower and upper large deviations spectra LD and LD are given by where the supremum is taken over all families of disjoint closed balls B i = B(x i , r) of radius r with centers x i in supp(µ).
It is clear that since µ is bounded, Equivalent definitions are (see [9]): One always has (see [37,29]) . and (see [2]) Definition 2.5. (Multifractal formalism) We say that µ obeys the multifractal formalism , and that the multifractal formalism holds (globally) for µ if it holds at any d ∈ R ∪ {∞} (here a negative dimension means that the set is empty).
Let us put our result in perspective with respect to the existing literature.
The study achieved in [41,42] leads to the multifractal nature of Gibbs measures projected on some random Cantor sets whose construction assumes a strong separation condition for the pieces of the construction. About the same time, the multifractal analysis of random Gibbs measures and Birkhoff averages on random Cantor sets and the whole torus were obtained in [23,24]; when the support of the measure is a Cantor set, a strong separation condition is assumed as well. More recently, in [15], the multifractal analysis for disintegrations of Gibbs measures on {1, . . . , m} N × {1, . . . , m} N was achieved as a consequence of the multifractal analysis of conditional Birkhoff averages of random continuous potentials (not C α ). The approach developed there could, with some effort, be adapted to derive our results on weak Gibbs measures if we worked with random fullshift only. However, as we already said it in the beginning of the introduction, the method cannot be extended easily to the random subshift, and our view point will be different. In [15], the authors start by establishing large deviations results, and then use them to construct by concatenation Moran sets of arbitrary large dimension in the level sets E(µ ω , d); we will concatenate information provided by random Gibbs measures associated with Hölder potentials which approximate the continuous potentials associated with the random weak Gibbs measure and the random maps generating the attractor X ω . This will provide us with a very flexible tool from which, for instance, we will deduce the result about the sets E(µ ω , d, d ). In this sense, our results also complete a part of those obtained in [35] which, in particular, achieves the multifractal analysis of random Gibbs measures on random Cantor sets obtained as the repeller of random conformal maps.
The multifractal analysis of Birkhoff averages on random conformal repellers of C 1 expanding maps is studied in [46], where the random dynamics is in fact coded by a non random subshift of finite type, and the random potentials that are considered satisfy an equicontinuity property stronger than the one we require.
The sets E(µ, d, d ) were studied for deterministic Gibbs measures on conformal repellers and for self-similar measures in [16,38,1,39].
(4) Equation (3) holds. Now, we can define Nevertheless there is a difference in the estimation of the local dimensions of measures.
As will see, in this paper, a building block in our proofs is the comparison of the mass assigned by a random Gibbs measure to neighboring basic intervals of the form U v ω , in order to control the mass assigned to centered intervals by a random weak Gibbs measure, as well as some auxiliary measures obtained by concatenation of pieces of random Gibbs measures (this point of view is fruitful in the study of the discrete inverses of random weak Gibbs measures in the companion paper mentioned at the beginning of this introduction).
In higher dimension the situation is different in general. If a strong separation condition is satisfied by the basic sets U v ω , there is no much difference with the 1 dimensional case. Otherwise, one can adapt the method used in [40] for self-conformal measures and under the open set condition, which, for any Gibbs measure ν, consists in controlling the asymptotic behavior of the distance of ν-almost every point x to the boundary of the basic set of the n-th generation containing x. We omit the details.

2.4.
Examples of random attractor. We end this section with examples illustrating our assumptions on the random attractors considered in this paper. As a first example, one has the fibers of McMullen-Bedford self-affine carpets, and more generally the Gatzouras-Lalley self-affine carpets [28], which naturally illustrate the idea that at a given step of the construction two consecutive intervals U s ω and U s+1 ω may touch each other. In [30], Luzia considers a class of expanding maps of the 2-torus of the form f (x, y) = (a(x, y), b(y)) that are C 2 -perturbations of Gatzouras-Lalley carpets, whose fibers illustrate our purpose with nonlinear maps. These examples are associated with random fullshift. Let us give a first more explicit example associated with a random subshift and a piecewise linear random maps.
Also, let σ be the shift map on Ω. Such a system is ergodic. It satisfies the conditions we need. For n = n 1 n 2 · · · n k · · · ∈ Γ, define l(n) = n 1 and A(n) the n 1 ×n 2 -matrix with all entries equal to 1 if n 2 = n 2 − 1 or n 1 = 2, and the n 1 × n 2 -matrix whose n 1 − 1 first rows have entries equal to 1 and the entries of the n 1 -th row equal 0 except that A n 1 ,n 1 −1 (n) = 1. It it is easy to check that both l and A are measurable, that log l dP < +∞, and l and A define a random subshift, which is not a fullshift. Also, the integer M = inf{m ∈ N : Notice that both l and M are unbounded.
Then we set T i n (x) = n 1 x mod 1 for x ∈ [ i−1 n 1 , i n 1 ] and for i = 1, 2, · · · , n 1 . In fact, the measure P defined above is a special example of a Gibbs measure on (Ω, F, σ) (see [44,45]). So we can enrich the previous construction by considering any such measure P for which log l dP < +∞. For the mappings maps T s ω , here is a way to provide a non trivial example, which seems to be not covered by the existing literature.
Start with a family {ϕ s,ω } s∈N of random C 1 differeomorphisms of [0, 1] such that at least one ϕ s,ω is nowhere C with positive probability. Assume that there exists a random variable a 0 taking values in (0, 1] and such that Then, the constant c ψ of equation (3) satisfies Thus, we require that This allows some T s ω be not uniformly expanding, but ensures expansiveness in the mean. It is easily seen that the Lebesgue measure of X ω is almost surely bounded by Thus, if we strengthen our requirement by assuming that then the Lebesgue measure of X ω is 0 almost surely. Now let us provide a completely explicit illustration of the last idea (we will work with a random fullshift for simplicity of the exposition).
We take (Ω, F, P, σ) as the fullshift It is the unique ergodic measure of maximal entropy for the shift map.
Let l be a random variable depending on ω 0 only, which is given by The entries of the random transition matrix are always 1 (we consider the random fullshift). We assume that the map T (ω, x) just depends on ω 0 and x. If In this case, we know that a 0 (ω) = 1; notice that the intervals In this case we can choose a 0 (ω) = 1/2. It is easy to check that T 1 ω is not expanding on some interval; furthermore it is just of class C 1 since h is nowhere -Hölder for any ∈ (0, 1). If It is easily checked that the left derivative of T 1 ω and the right derivative of T 2 ω do not coincide, so the dynamics is not the restriction of a random conformal map. In this case we can choose a 0 (ω) = 7/8. Also, so that all the conditions hold.
3. Basic properties of random weak Gibbs and random Gibbs measures. Approximation of (Φ, Ψ) by random Hölder potentials This section prepares the proofs of our main results. Sections 3.1 and 3.2 present basics facts about random weak Gibbs and random Gibbs measures. Section 3.3.1 provides an approximation of (Φ, Ψ) by a family {(Φ i , Ψ i )} i≥1 of random Hölder potentials. Section 3.3.2 derives some related properties of the associated pressure functions, which yield the variational formulas appearing in theorem 2.6. Section 3.3 presents properties related to the random Gibbs measures associated with the couple (Φ i , Ψ i ), which will be used as building blocks in the concatenation of measures used in the proof of the main parts of theorem 2.6 and of theorem 2.7 (section 4).
Furthermore, the variation Principle holds, which means, is defined as in proposition 1. The following lemma is direct when the potential Φ possesses bounded distorsions so that the Ruelle-Perron-Frobenious theorem holds for the operator L ω Φ . For general potentials in L 1 Σ Ω (Ω, C(Σ)) we need a proof. Lemma 3.3. One has lim n→∞ log λ(ω, n) n = P (Φ) for P-almost every ω ∈ Ω.
Remark 2. In the next proof, as well as in the rest of the paper, we will use the letter M to denote the levels of the function M (·). Keeping this in mind should prevent from some confusion.
. The second inequality uses the fact that we work with a subshift as well as (10). We just prove the first inequality: for n large enough so that M (σ n ω) ≤ n, By using the topological mixing property and preserving for each w ∈ Σ ω,n−M (σ n ω) only one path of length M (σ n ω) from w to v, the inequality follows from (10), M (σ n ω) = o(n) and (11). Now, since λ(ω, n) = L ω,n Φ 1(v)d µ σ n ω (v), we can easily get the result from lemma 3.2 and the fact that M (σ n ω) = o(n).
Recall that by proposition 1, for P-a.e ω ∈ Ω, the measures µ ω satisfy for any v ∈ [v] ω , where n does not depend on v and tends to 0 as n → ∞.
Proof. Let us deal first with the case n = 1. .
Then, taking the sum over Finally we conclude with (10) and log D(σ n ω) = o(n).
For any γ ∈ L 1 X Ω (Ω, ‹ C([0, 1])) and any z ∈ U v ω , let From the Lagrange's finite-increment theorem, distortions and proposition 2, using standard estimates we can get the following proposition.
Each time we need to refer to the function Φ, we denote the measures m and m ω as m Φ and m Φ ω , and denote λ as λ Φ . We can also define a random Gibbs measure on the random attractor X ω by setting µ ω = m ω • π −1 ω . Given a random Hölder potential Φ, from (13) we can define the normalized potential , which satisfies L ω Φ 1 = 1 for P-almost every ω ∈ Ω. This implies that Φ ≤ 0 for P-almost every ω ∈ Ω. Also, we have the following fact: Proposition 4. Suppose that Φ is a random Hölder potential. If P (Φ) = 0, there exist some > 0 such that for P-almost every ω ∈ Ω, there exists N (ω) such that for any n ≥ N (ω) and any v ∈ Σ ω,n , one has As a consequence, µ ω is atomless.
If we need to refer explicitly to Φ, we will use the notations N Φ (ω) and Φ instead of N (ω) and .
The main idea of the proof is from [15].
3.3. Approximation of (Φ, Ψ) by random Hölder potentials, and related properties. We mainly introduce objects and related properties which will be used in the following sections. Also, we explain the variational formulas appearing in the statement of theorem 2.6.
3.3.1. Approximation of (Φ, Ψ) by random Hölder potentials. Now we approximate the potentials Φ and Ψ associated with {µ ω } ω∈Ω and {X ω } ω∈Ω by more regular potentials: for any i ≥ 1, for any ω ∈ Ω for any These functions Φ i and Ψ i are piecewise constant with respect to the second variable. They are random Hölder continuous potentials. If we take and the right hand side is integrable since Φ ∈ L 1 Σ Ω (Ω, C(Σ)). Also, since for P-almost every ω we have var n Φ(ω) → 0 as n → +∞, and Φ(ω) − Φ i (ω) ∞ ≤ var i Φ(ω), we have Φ i → Φ uniformly as i → ∞ for P-almost every ω. The same property holds for Ψ i and Ψ. Consequently, without loss of generality we can also assume that P (Φ i ) = 0 since P (Φ i ) converges to P (Φ) as i tends to +∞.

3.3.2.
Approximation of (T, T * ) by (T i , T * i ). Due to our assumptions on (Φ, Ψ) and the definition of (Φ i , Ψ i ) i∈N , we have c Ψ i > 0, hence for the same reason as for (Φ, Ψ), for any q ∈ R, for any i ∈ N, there exists a unique T i (q) such that P (qΦ i − T i (q)Ψ i ) = 0 and the function T i is concave and non-decreasing. Also, the function T i is differentiable since for Hölder potentials the associated random Gibbs measure is the unique invariant measure that maximizes the variation principle (see [18,26,35].) Lemma 3.5. For any q ∈ R, one has that T i (q) → T (q) as i → ∞.
Lemma 3.6. Let ‹ T : R → R be a concave function. Suppose that ( ‹ T i ) i≥1 is a sequence of differentiable concave functions from R to R which converges pointwise to ‹ T . Then ( ‹ T * i ) i≥1 converges pointwise to ‹ T * over the interior of the domain of ‹ T * .
Proof. Let α be an interior point of dom( ‹ T * ). Let q α ∈ R be the unique point such that α ∈ [ ‹ T (q α +), ‹ T (q α −)], and ‹ T * (α) = αq α − ‹ T (q α ). By [11, proposition 2.5(i)], there exists a sequence (q i ) i≥1 such that for i large enough one has ‹ T i (q i ) = α. Without loss of generality we can assume that this sequence converges to q 0 ∈ R or diverges to −∞ or ∞.
Suppose first that it converges to q 0 ∈ R. If q 0 = q α then we are done since ( ‹ T i ) i≥1 converges uniformly on compact sets. Suppose that q 0 = q α and q 0 > q α . Using the uniform convergence of ( ‹ T i ) i≥1 in a compact neighborhood of [q α , q 0 ] and the inequality

On the other hand, T being concave we have
). The case q 0 = q α and q 0 < q α is similar. Now suppose that (q i ) i≥1 diverges to ∞ (the case where it diverges to −∞ is similar). If ‹ T is affine over [q α , ∞) with slope α, α is not an interior point of dom( ‹ T * ). Consequently, there exists q 0 and > 0 such that ‹ T (q 0 +) < α − , and ‹ T (q) ≤ ‹ T (q 0 ) + (α − )(q − q 0 ) for all q ≥ q 0 . On the other hand, since ‹ T i is non increasing for all i, for i large enough we have

3.3.3.
Explanation of some variational formulas in theorem 2.6.  Proof. The proof of (16) is just use the fact of the variation principle, see (12). Regarding the equation (17), on the one hand, for any d ∈ R, On the other hand, for any d ∈ (T (+∞), T (−∞)), by the proof of lemma 3.6, there exists i large enough and q i ∈ R such that T i (q i ) = d and P This implies that there exists ρ i ∈ I P (Σ Ω ) such that Lemma 3.6 tells us that T * i (d) → T * (d) as i → ∞ and I P (Σ Ω ) is compact for the weak* topology. Thus, there exists a limit point ρ of (ρ i ) in I P (Σ Ω ) such that of elements of (T (+∞), T (−∞)) and for each k picking ρ k which realizes max Then, since T * is continuous at d (it is lower semi-continuous as a concave function and upper semi-continuous as a Legendre transform), any limit point of (ρ k ) k≥0 is such that Φdρ Ψdρ = d. It exists since I P (Σ Ω ) is compact in the weak* topology (see [25,27]).

3.3.4.
Simultaneous control for random Gibbs measures associated with (Φ i , Ψ i ). In this quite technical subsection, we prepare the "concatenation of random Gibbs measures" approach that will be used in the next sections to construct auxiliary measures with nice properties. We also show an almost everywhere almost doubling property for the random Gibbs measures on the random attractor X ω . Let D be a dense and countable subset of (T (+∞), T (−∞)). Let {D i } i∈N be an increasing sequence of finite sets such that ∪ i∈N D i = D.
Fix a sequence {ε i } i∈N decreasing to 0 as i → ∞. Due to lemma 3.5 and the proof of Lemma 3.6, for any i ∈ N there exists j i large enough such that for any d ∈ D i , there exists q i = q i (d) ∈ R such that the following properties hold: We can also assume that j i+1 > j i for each i ∈ N. We set We also define var i Φ(ω) := var j i Φ(ω).
Then for any i ∈ N, from the ergodic theorem, for P-almost every ω lim s→∞ θ (i, ω, s) s = 1

P(Ω(i))
. Consequently, Since N is countable, there exists ‹ Ω ⊂ Ω of full probability such that for all ω ∈ ‹ Ω , for any i ∈ N, we have Given ω ∈ Ω(i), let where s k is the smallest s such that the following property holds: It is easily seen that Now we prove an almost everywhere almost doubling property for the Gibbs measures µ For v ∈ Σ σ M (i) ω,n , we denote by U v+ σ M (i) ω and U v− σ M (i) ω the two intervals of the n-th generation of the construction of X ω which are neighboring U v σ M (i) ω , whenever U v σ M (i) ω is neither the leftmost nor the rightmost of the whole collection, and with the convention Proof. For any v and v such that |v| = n i k−1 , |v | = n i k − n i k−1 and vv ∈ Σ σ M (i) ω,n i k , by We will use this fact to estimate the measure of U(i, σ M (i) ω, k). Notice that for any v ∈ Σ σ M (i) ω,n i k−1 , there are at most two v such that vv ∈ Σ σ M (i) ω,n i k and B(i, σ M (i) ω, k, vv ) holds. Consequently, .
By Borel-Cantelli's lemma we get µ For any ε > 0, β ≥ 0, and k, p ≥ 1 we now define the following sets: and then Lemma 3.9. For all i ∈ N, for any ε > 0, for all ω ∈ Ω(i), for all q ∈ Q i , the singularity We have Since T i is in fact not only differentiable, but analytic [18,35], we have uniformly in q ∈ Q i . Thus, there exists b > 0 such that for η small enough, for all q ∈ Q i , we have Consider such an η in (0, ε 2b ]. We have Consequently, +∞ k=1 S i,q,k < ∞, which by the Borel-Cantelli lemma yields the desired conclusion since ε is arbitrary. Now we can collect the following facts. In fact with a suitable change of ε i (take it as 2ε i ), we can get the following additional properties from (18), (19), and (20) above: Fact 2. We can change Ω(i) to Ω i ⊂ Ω(i) a bit smaller such that P(Ω i ) ≥ 1 − and there exist N i and W (i) such that for any ω ∈ Ω i , N i (σ M (i) ω) ≤ N i and n i N i (ω) ≤ W (i) and the properties listed in Facts 1 hold.
We define θ(i, ω, s) as being the s-th return time to the set Ω i for the point ω. Since N is countable, there exists ‹ Ω ⊂ ‹ Ω of full probability such that for all ω ∈ ‹ Ω, for any i ∈ N, we have

Multifractal analysis of random weak Gibbs measures:
Proof of theorems 2.6 and 2.7 This section consists of three subsections. In the first one we obtain the sharp lower bound for the lower L q -spectrum of µ ω . Next, in the second subsection, we prove the validity of the mutifractal formalism (see theorem 2.6(2)). Due to (8), (9) and lemma 3.7, we just need to prove dim H E(µ ω , d) = τ * µω (d). There, our approach to construct suitable auxiliary measures already prepares the material used to establish in the third subsection the refinements gathered in theorem 2.6(3)(4) and theorem 2.7.
We will establish the lower bound τ µω (q) ≥ T (q) for all ω ∈ " Ω and q ∈ D. Since D is dense and both τ µω and T are continuous, this will yield τ µω ≥ T for all ω ∈ " Ω. By using the multifractal formalism, this immediately yields the desired upper bound T * for τ * µω and the various spectra we consider for µ ω . The equality τ µω = τ µω then follows from standard considerations in large deviations theory.
Fix ω ∈ " Ω. Let r > 0 and consider B = {B i }, a packing of X ω by disjoint balls B i with the center x i and radius r. For each ball B i , choose n = n i and v(x i ) ∈ Σ ω,n such that By removing a set of probability 0 from " Ω if necessary, for any v ∈ [v(x i )] ω , we have where we have used ergodic theorem. Thus n ≥ − log r 2C Ψ for r small enough. On the other hand, for r small enough, for any where we have applied proposition 3(2) to the potential qΦ − T (q)Ψ as well as proposition 3(1), the fact that |U (24). It follows that i (µ ω (B i )) q ≤ r T (q) exp(o(− log r)), and this bound does not depend on the choice of the packing {B i }. Letting r → 0 yields τ µω (q) ≥ T (q).
For any v ∈ V (ω, r), U v ω meets at most exp(o(− log r)) many balls of B i and for any Using the same argument as for q < 0, we can know get that n (ω,r)≤n≤n(ω,r) v∈Σω,n∩V (ω,r)

4.2.
Lower bound for the Hausdorff spectrum. Recall facts 1 and facts 2 derived at the end of section 3.3.4. For any ω ∈ ‹ Ω, for any d ∈ [T (+∞), T (−∞)], for any sequence {d i } i∈N with d i ∈ D i , such that lim i→∞ d i = d, and consequently lim i→∞ T * (d i ) = T * (d) by continuity of T * , we will construct a probability measure η ω supported on a set This will imply that dim H η ω ≥ T * (d), and then For each i ∈ N, Facts 1 will be applied with this i .
From now on: • we only deal with points ω in the set ‹ Ω of P-probability 1 for which the sequence {θ(i, ω, s)} i,s∈N,ω∈ Ω is well defined; • the sequence {Ω i , N i , W (i), ε i } i∈N is fixed; • for each ω ∈ Ω i the sequence {n i k } k∈N is well defined; • we denote the properties listed in fact 1 by GP (ω, i, q). More precisely, we denote the five items by GP (ω, i, q)(1) to GP (ω, i, q)(5). We will build a family of Moran structures indexed by the elements of i≥1 D i . For any ω ∈ ‹ Ω, recall that θ(1, ω, 1) is the smallest n ∈ N such that σ n ω ∈ Ω 1 ⊂ Ω(1). Define m 1 := θ(1, ω, 1) + M (1). Facts 1 and facts 2 tell us that for any d 1 ∈ D 1 , there exists q 1 ∈ Q 1 such that T 1 (q 1 ) = d 1 and a set E 1, • for any s such that the return time θ(2, ω, s) satisfies θ(2, ω, s) ≥ m 1 + n 1 Let s 2 be the smallest s such that θ(2, ω, s) ≥ m 1 + n 1 N 1 . Now, let N 1 be the largest k such that m 1 + n 1 k ≤ θ(2, ω, s 2 ) (by construction we have Here is a picture which illustrates the beginning of the construction. For each q ∈ Q 1 and k ≥ 1, let (1,ω,1) . For any d 1 ∈ D 1 , we define: (recall that the meaning of s * s is specified at the beginning of section 2.1).
Then choose N i+1 large enough such that . The above two items ensure that we do not need to wait a long relative time to go to the next step.
Remark 3. By construction, we can take n i+1 as big as we want (In the construction ). This implies that we can get the sequence {m i } i∈N increasing as fast as we want. We can also impose that ). Thus, the speed we fix for the growth of (n i N i ) i∈N directly impacts the growth speed of (m i ) i∈N .
Here again we draw a picture to illustrate this construction.
As in the case i = 0, for any d i+1 ∈ D i+1 , we can define For any d ∈ [T (+∞), T (−∞)], there exists {d i } i∈N ∈ i∈N D i , such that lim i→∞ d i = d and lim i→∞ T * (d i ) = T * (d). Moreover, if d ∈ (T (+∞), T (−∞)), T * i (d i ) converges to T * (d) directly from lemma 3.6. If d ∈ {T (+∞), T (−∞)}, again due to lemma 3.6, we can choose (d i ) i≥1 to be piecewise constant to make sure that We fix such a sequence and suppose that the correspondence q are {q i } i∈N . Define We will prove that for any x ∈ K(ω, {d i } i≥1 ), one has lim r→0+ log µω(B(x,r)) log r = d. Then To do so, we first establish two general estimates: First estimate. For any w ∈ R i , v ∈ V(σ m i+1 ω, i+1, q i+1 , k), for any k ≥ N i+1 , for any v ∈ [w * v] ω for Υ ∈ {Φ, Ψ} we have (remembering the notations introduced at the beginning of section 3.3.4 and the fact that by construction we have |Υ(ω, v)−Υ p (ω, v)| ≤ var p Υ(ω)): for i and k large enough, where in the last inequality we have used the fact that m i ≤ m i+1 ε 3 i+1 and ε i ≥ ε i+1 > 0.
Second estimate. For i ∈ N large enough, for any k with Indeed, for i large enough, we have We can now estimate the local dimension of µ ω . Fix x ∈ K(ω, {d i } i≥1 ). If r is small enough, we can choose the largest i, then the largest k = k i+1 , with N i+1 < k ≤ N i+1 , such that the following property holds: there exists w ∈ R i (d 1 , From the construction, if U w * v+ ω and U w * v− ω are the neighboring intervals of U w * v ω , then |v ∧ v + |, |v ∧ v − | are larger than n i+1 k−1 . Then by (27) we have |U w * v+ ω | ≥ 2r and Now, using estimates similar to those leading to (27) with Ψ replaced by Φ we get that for any w * v ∈ [w * v] ω , Consequently, using (26), Now let us estimate log r from below: Observe that n i+1 (29) and (26) we can get Consequently, We have where we have used (25). This implies that (30) holds as well.
Finally, for any v ∈ (w * v) ω , (28) and (30) imply Due to the construction and see (21), we have | It follows from Stolz-Cesàro theorem that It remains to prove that lim sup r→0 log µω(B(x,r)) log r ≤ d. This is easier than for the lim inf since we just have to choose the smallest i and then the smallest In both cases we have Finally we have which yields lim sup r→0 log(µ ω (B(x, r))) log r ≤ d. .
We can extend η ω in a unique way to a probability measure on the σ-algebra generated by 2 , using the same approach as the proof of (33), we can get that for any x ∈ K(ω, {d i } i≥1 ), This yields dim H (E(µ ω , d)) ≥ T * (d).

Remark 4.
In fact, if in the construction the sequence (m i ) i∈N is replaced by another one growing faster (with the effect to modify (K(ω, lim inf  This property will be used in the next subsection.
Proof of (38) and ( where for the first inequality we have used the same estimates as to get (32). Also, if we assume that m i−1 ≤ m i ε 3 i . Thus equation (38) is established. To get (39), observe that lim sup r→0 log η ω (B(x, r)) log r if we assume that m i−1 ≤ m i ε 3 i . On the other hand,
Proof of theorem 2.6(3). We first deal with the lower bounds for the dimensions. Setd 2i = d i andd 2i+1 = d i . We can use the same construction as in the previous subsection and get a set K(ω, {d i } i≥1 ) ⊂ E(µ ω , d, d ), as well as a probability measure η ω supported on K(ω, {d i } i≥1 ).
Since T * (d) < t 0 , we can find {d i } i∈N ∈ i∈N D i such that d i → d as i → ∞, T * (d i ) − ε i ≥ T * (d) + ε i for i large enough, T * (d i ) → T * (d) as i → ∞, and T * (d i ) − ε i is ultimately non increasing.
For any gauge function g such that lim sup r→0 log g(r) log r ≤ T * (d), there exists a positive sequence {υ r } r>0 such that both υ r and r υr decrease to 0 as r decreases to 0 and g(r) ≥ r T * (d)+υr (r ≤ 1).
Finally, suppose that g is a gauge function such that lim inf r→0 log g(r) log r ≤ T * (d). There exist {r j } i∈N ∈ (0, 1) N , and {υ r j } j∈N ∈ (0, ∞) N such that υ r j ∈ (0, 1] and r υ j j decrease to 0 as j tends to ∞, and g(2r j ) ≥ r T * (d)+υr j j . Using the same approach as above, we can choose (d i ) ≥1 ∈ i≥1 D i such that lim i→∞ d i = d, T * (d i ) converges slowly to T * (d) from above, and in the construction of (K(ω, {d i } i≥1 ), η ω ), m i tends fast enough to ∞ so that, for some j 0 ∈ N, for all j ≥ j 0 , for any x ∈ K(ω, {d i } i≥1 , η ω (B(x, r j )) ≤ C (2r j ) T * (d)+2υr j . Now, let A ⊂ K(ω, {d i } i≥1 ) be of positive η ω -measure. For any given δ > 0, take j 0 ≥ j 0 such that r j 0 ≤ δ consider the following family of closed balls which is a covering of A. Due to Besicovitch covering theorem, we can extract an at most countable subfamily of pairwise disjoint balls {B(x i , ρ i )} i∈I such that η ω ( i∈I B i ) > 0.
This family is a δ-packing of A, and (P g 0,δ stands for the prepacking measure associated with g) As j 0 → ∞ when δ → 0, we can conclude that P g 0 (A) = +∞. Then, since any at most countable covering of K(ω, {d i } i≥1 ) must contain a set A of positive η ω -measure, we finally get P g (K(ω, {d i } i≥1 )) = +∞, so P g (E(µ ω , d)) = +∞.