Julien Barral

S T Éphane Seuret

THE FRISCH-PARISI CONJECTURE I: PRESCRIBED MULTIFRACTAL BEHAVIOR, AND A PARTIAL SOLUTION

, we introduce and study a family B = {B µ,p q (R d )} µ∈E d ,(p,q)∈[1,+∞] 2 of heterogeneous Besov spaces that contains {B µ (R d )} µ∈E d and generalises in a natural direction the family of standard Besov spaces, and we solve the inverse problem exhaustively inside B.

Introduction

This paper and its companion [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF] deal with multifractal analysis of functions, which originates from the first geometric quantification of the Hölder singularities structure in fully developed turbulence [START_REF] Mandelbrot | Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées[END_REF][START_REF] Mandelbrot | Intermittent turbulence in self-similar cascades, divergence of high moments and dimension of the carrier[END_REF][START_REF] Frisch | Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics[END_REF]. This subject is an instance of the natural concept of multifractality, which comes into play as soon as, given a mapping h : X → A between a metric space (X, d) and a set A, one wants to describe geometrically the level sets of h by considering the mapping σ : α ∈ A → dim h -1 ({α}), where dim stands for the Hausdorff dimension. Indeed, in many situations, the level sets of h form an uncountable family of (disjoint) fractal sets, and σ is sometimes called multifractal spectrum. This spectrum provides a hierarchy between these level sets, according to their size measured by their Hausdorff dimension. Such spectra occur in many mathematical fields, such as harmonic and functional analysis (in the description of fine properties of Fourier series [START_REF] Jaffard | The spectrum of singularities of Riemann's function[END_REF][START_REF] Bayart | Multifractal analysis of the divergence of Fourier series[END_REF] or typical elements in function spaces [START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF][START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF]), probability theory (to describe fine properties of Brownian motion or SLE curves [START_REF] Orey | On the Hausdorff dimension of brownian slow points[END_REF][START_REF] Perkins | On the Hausdorff dimension of Brownian slow points[END_REF][START_REF] Lawler | Almost sure multifractal spectrum for the tip of an SLE curve[END_REF][START_REF] Gwynne | Almost sure multifractal spectrum of schramm-loewner evolution[END_REF], multiplicative chaos and Gaussian free field, random covering problems [START_REF] Barral | Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures[END_REF][START_REF] Hu | Thick points of gaussian free fields[END_REF][START_REF] Rhodes | Gaussian multiplicative chaos and applications: A review[END_REF][START_REF] Barral | Covering numbers of different points in the Dvoretzky covering[END_REF]), ergodic theory, dynamical and iterated function systems (to analyse Gibbs/harmonic measures on conformal repellers, Birkhoff averages, and self-similar measures [START_REF] Pesin | The multifractal analysis of Gibbs measures: Motivation, mathematical foundation, and examples[END_REF][START_REF] Makarov | Fine structure of harmonic measure[END_REF][START_REF] Feng | Gibbs properties of self-conformal measures and the multifractal formalism[END_REF][START_REF] Feng | Multifractal formalism for self-similar measures with weak separation condition[END_REF][START_REF] Shmerkin | On Furstenberg's intersection conjecture, self-similar measures, and the l q norms of convolutions[END_REF]), metric number theory (Diophantine approximation and ubiquity theory [START_REF] Jarnik | Diophantischen approximationen und Hausdorffsches mass[END_REF][START_REF] Hill | The Jarnìk-Besicovitch theorem for geometrically finite Kleinian groups[END_REF][START_REF] Barral | Ubiquity and large intersections properties under digit frequencies constraints[END_REF], shrinking targets problems and dynamical covering problems [START_REF] Hill | The ergodic theory of shrinking targets[END_REF][START_REF] Fan | A multifractal mass transference principle for Gibbs measures with applications to dynamical diophantine approximation[END_REF]), the previous references being far from exhaustive.

In the multifractal analysis of a real valued function f ∈ L ∞ loc (R d ), the function h of interest is the pointwise Hölder exponent function h f , which is defined as follows. Given x 0 ∈ R d , and H ∈ R + , f is said to belong to C H (x 0 ) if there exist a polynomial P of degree at most H , a constant C > 0, and a neighborhood V of x 0 such that

∀ x ∈ V, |f (x) -P (x -x 0 )| ≤ C|x -x 0 | H .
Date: Received: date / Revised version: date.

The pointwise Hölder exponent

of f ∈ L ∞ loc (R d ) at x 0 is (1.1) h f (x 0 ) = sup ¶ H ∈ R + : f ∈ C H (x 0 ) © ,
and f is said to have a Hölder singularity of order h f (x 0 ) at x 0 . The associated multifractal spectrum, also called singularity spectrum of f , is the mapping

σ f : H ∈ R ∪ {∞} → dim E f (H) ∈ [0, d] ∪ {-∞}
, where E f (H) := h -1 f ({H}) (note that E f (H) = ∅ for H < 0). Again, dim stands for the Hausdorff dimension, with the convention dim ∅ = -∞. The function f is said to be multifractal when E f (H) = ∅ for at least two values of H.

The idea of considering this spectrum goes back to the physicists U. Frisch and G. Parisi [START_REF] Frisch | Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics[END_REF], who aimed at quantifying geometrically the local variations of the velocity field of a turbulent fluid, and introduced the term multifractal. Another fundamental idea pointed out by Frisch and Parisi consisted in coupling the singularity spectrum with a large deviations approach, in order to statistically describe the Hölder singularities distribution (similar to Mandelbrot's approach for measures [START_REF] Mandelbrot | Multiplications aléatoires itérées et distributions invariantes par moyennes pondérées[END_REF]). This led to the notion of multifractal formalisms for functions. In such a formalism, the singularity spectrum σ f of a Hölder continuous function f is always dominated by (and in good cases, coincides with) the Legendre-Fenchel transform

ζ * f (H) := inf q∈R Hq -ζ f (q)
of a mapping ζ f : R → R, called the scaling function or the L q -spectrum of f : for every h ≥ 0, σ f (h) ≤ ζ * f (h). This mapping ζ f is a kind of free energy function encapsulating the asymptotic statistical distribution of the Hölder singularities as the observation scale tends to 0, and it can be numerically estimated [START_REF] Jaffard | Wavelet leaders in multifractal analysis[END_REF]. For instance, in their seminal article, Frisch and Parisi used for ζ f the scaling exponent of the moments of the increments of f , informally defined as

|h| -d Ω |f (x + h) -f (x)| q dx ∼ |h| ζ f (q) as h → 0,
where Ω is a fixed bounded domain on which f is supposed to be fully supported. The heuristics developed in [START_REF] Frisch | Fully developed turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics[END_REF] lead to seek for the largest classes of functions for which the equality

(1.2) σ f (H) = ζ * f (H) holds at any H such that ζ * f (H) ≥ 0.
In such a situation, one says that the multifractal formalism holds for f , or that f satisfies the multifractal formalism. Then, the spectrum σ f is a continuous concave mapping with support included in (0, +∞), and assuming that the topological support of f is full, one necessarily has σ f (H) = d = -ζ f (0) for some H ≥ 0 (for instance the level set E f (H) may have a positive Lebesgue measure).

We will come back to rigorous definitions of multifractal formalisms for measures in Section 2.2, and for functions in the second paper [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF]. The concept of multifractal formalism motivated many works in geometric measure theory [START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Olsen | A multifractal formalism[END_REF][START_REF] Lau | Multifractal measures and a weak separation condition[END_REF][START_REF] Véhel | Multifractal analysis of Choquet capacities[END_REF], dynamical systems in connection with the thermodynamic formalism [START_REF] Pesin | Dimension theory in dynamical systems[END_REF], and analysis [START_REF] Jaffard | Multifractal formalism for functions, Part 1: Results valid for all functions, Part 2: Selfsimilar functions[END_REF][START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF][START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF]. It provides a powerful framework to describe the fine geometric structure of invariant measures of various dynamical systems [START_REF] Collet | The dimension spectrum of some dynamical systems[END_REF][START_REF] Rand | The singularity spectrum f (α) for cookie-cutters[END_REF][START_REF] Pesin | Dimension theory in dynamical systems[END_REF] and self-similar and self-affine measures [START_REF] King | The singularity spectrum for general Sierpinski carpets[END_REF][START_REF] Olsen | A multifractal formalism[END_REF][START_REF] Olsen | Self-affine multifractal Sierpinski sponges in R d[END_REF][START_REF] Lau | Multifractal measures and a weak separation condition[END_REF][START_REF] Feng | Multifractal formalism for self-similar measures with weak separation condition[END_REF][START_REF] Barral | Multifractal formalism for almost all self-affine measures[END_REF], self-similar functions [START_REF] Jaffard | Multifractal formalism for functions, Part 1: Results valid for all functions, Part 2: Selfsimilar functions[END_REF], as well as limit measures or functions in multiplicative chaos theory [START_REF] Holley | Multifractal dimensions and scaling exponents for strongly bounded random fractals[END_REF][START_REF] Barral | Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures[END_REF][START_REF] Barral | Multifractal analysis of complex random cascades[END_REF]. The singularity spectrum and its suitable extensions to non-bounded functions have also been used to describe the regularity properties of celebrated functions like Riemann's and Brjuno's functions [START_REF] Jaffard | The spectrum of singularities of Riemann's function[END_REF][START_REF] Seuret | Local L 2 -regularity of Riemann's Fourier series[END_REF][START_REF] Jaffard | Multifractal analysis of the Brjuno function[END_REF], stochastic processes like Lévy processes and general classes of Markov processes [START_REF] Jaffard | The multifractal nature of Lévy processes[END_REF][START_REF] Barral | A pure jump Markov process with a random singularity spectrum[END_REF][START_REF] Yang | Multifractality of jump diffusion processes[END_REF], as well as Lévy processes in multifractal time [START_REF] Barral | The singularity spectrum of Lévy processes in multifractal time[END_REF]. Coming back to turbulence, it is worth mentioning that recently Banica and Vega established a remarkable connection between multifractal functions obeying some multifractal formalism, namely Riemann's function n≥1

sin(n 2 x) n 2
and some of its variants, and specific solutions to the binormal curvature flow, a geometric PDE used to model the evolution of vortex filaments [START_REF] Banica | Riemann's non-differentiable function and the binormal curvature flow[END_REF]. Multifractal formalisms are also relevant in many other applications, due to the existence of stable algorithms that precisely estimate scaling functions ζ f of numerical data. Then, a key observation is that for most of real-life data associated with intermittent phenomena, their estimated singularity spectra ζ * f have a characteristic strictly concave bell shape (see [START_REF] Abry | Irregularities and scaling in signal and image processing: Multifractal analysis[END_REF] and Figure 1). This is also the case for the singularity spectra of important classes of functions possessing scaling properties [START_REF] Jaffard | Multifractal formalism for functions, Part 1: Results valid for all functions, Part 2: Selfsimilar functions[END_REF][START_REF] Barral | The singularity spectrum of Lévy processes in multifractal time[END_REF][START_REF] Barral | Multifractal analysis of complex random cascades[END_REF]. This behavior is in striking contrast to the results established for typical functions in the classical function spaces, where "typical" is meant in the sense of Baire categories 1 . Indeed, it has been proved that typical increasing real functions (Buczolich&Nagy [START_REF] Buczolich | Hölder spectrum of typical monotone continuous functions[END_REF]), typical functions in Sobolev and Besov spaces (Jaffard [36], Jaffard&Meyer [START_REF] Jaffard | On the pointwise regularity of functions in critical Besov spaces[END_REF]), and typical measures (Buczolich&Seuret, Bayart [START_REF] Buczolich | Typical Borel measures on [0, 1] d satisfy a multifractal formalism[END_REF][START_REF] Bayart | Multifractal spectra of typical and prevalent measures[END_REF]) satisfy a multifractal formalism and have an affine increasing singularity spectrum. One concludes that, from the multifractal standpoint, realistic behaviors are not reproduced by typical elements in the standard function spaces. Precise statements regarding typical singularity spectrum and multifractal formalisms in Besov spaces are discussed in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF].

On the other hand, the previous genericity results show that many multifractal functions do satisfy some multifractal formalism without assuming any scale invariance properties. In [START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF], Jaffard seeks for Baire topological spaces of functions in which typical functions have a prescribed singularity spectrum, and do obey some multifractal formalism. He names this inverse problem "Frisch-Parisi conjecture", and provides a partial solution to it: some intersections of homogeneous Besov spaces are Baire topological spaces in which typical functions possess an increasing compactly supported singularity spectrum, with a prescribed concave part, and another affine part; moreover, typical elements partially obey some multifractal formalism (see [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF] for a detailed description of Jaffard's result). Again, no scale invariance is assumed. 1 Recall that in a Baire topological space E, a property P is called typical, or generic, when the set {f ∈ E : f satisfies P} is of second category in E, or equivalently contains a dense G δ -set, that is the intersection of a countable family of dense and open sets. One says that typical elements in E satisfy P when P is typical in E. Note that the set S d consists of those mappings σ which are admissible to be the singularity spectrum of some Hölder continuous function f : R d → R whose pointwise Hölder exponents range in a compact subinterval of (0, +∞), such that dim E f (H) = d for at least one exponent H, and which obeys a multifractal formalism.

In the present paper, we first prove the existence of almost-doubling and Z d -invariant Radon measures fully supported on R d with prescribed singularity spectrum, and which satisfy the standard multifractal formalisms for measures developed in [START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Olsen | A multifractal formalism[END_REF] (Theorem 4). Such a measure ν possesses scaling-like properties. Also, it possesses the homogeneous property that its restrictions to any non-empty closed ball of positive measure shares the same multifractal properties as ν. Up to now, such a property for measures obeying the multifractal formalism and with prescribed singularity spectrum was only known for measures supported on a totally disconnected set [START_REF]Inverse problems in multifractal analysis of measures[END_REF] (see also [START_REF] Buczolich | Measures and functions with prescribed singularity spectrum[END_REF] for results on the prescription of the singularity spectrum for measures). Then, we introduce a family {B µ (R d )} µ∈E d of Baire functions spaces indexed by the set E d of capacities µ obtained as positive powers of one of the measures ν built before (i.e. there exists s > 0 such that µ(E) = ν(E) s for all E ∈ B(R d )). The definition of B µ (R d ) is based on wavelets; an equivalent definition based on moduli of smoothness is achieved in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF]. We prove that Baire typical functions in B µ (R d ) have a multifractal spectrum equal to that of µ. Since prescribing the multifractal spectrum of measures makes it possible to prescribe that of capacities, this yields the following result: Theorem 2. For every σ ∈ S d , there exists a Baire space of functions f : R d → R in which any Baire typical element f satisfies σ f = σ. This does not solve completely the Frisch-Parisi conjecture, since the multifractal formalism is not investigated in this paper. This is achieved in the second paper [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF], where a much wider family {B µ,p q (R d )} µ∈E d ,(p,q)∈[1,+∞] 2 of Baire function spaces, extending classical Besov spaces and referred to as Besov spaces in multifractal environment, is introduced and studied (B µ (R d ) corresponds to the special case p = q = ∞). Using the spaces B µ,p q (R d ) with suitable parameters µ, p and q yields various solutions to the original inverse problem: Theorem 3 ( [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF]). Conjecture 1 is true.

Next section describes the steps leading to Theorem 2.

Statements of the main results

2.1. Some notations and definitions. The Lebesgue measure on

R d is denoted L d . If E is a Borel subset of R d , the Borel σ-algebra of E is denoted B(E).
|E| stands for the Euclidean diameter of E.

Given x ∈ R d and r ∈ R + , the closed Euclidean ball centered at x with radius r is denoted B(x, r).

For j ∈ Z, D j stands for the collection of closed dyadic cubes of generation j, i.e. the cubes λ j,k = 2 -j k + 2 -j [0, 1] d , where k ∈ Z d . We also set D = j∈Z D j , and if λ = λ j,k ∈ D j we denote x λ = 2 -j k.

For x ∈ R d , λ j (x) stands for the closure of the unique dyadic cube of generation j, product of semi-open to the right dyadic intervals, which contains x.

For j ∈ Z, λ ∈ D j , and N ∈ N * , N λ denotes the cube with same center as λ and radius equal to N • 2 -j-1 in (R d , ∞ ). For instance, 3λ is the union of those λ ∈ D j such that ∂λ ∩ ∂λ = ∅ (∂λ stands for the frontier of the cube λ).

The domain of a function g : R → R ∪ {-∞} is defined as g -1 (R), and denoted by dom(g). When g is concave and finite, one sets g (+∞) = lim t→+∞ g (t + ) and g (-∞) = lim t→-∞ g (t + ).

Definition 2.1. The set of Hölder set functions on B(R d ) is defined as

(2.1) H(R d ) = ¶ µ : B(R d ) → R + ∪ {∞} : ∃ C, s > 0, ∀ E ⊂ R d , µ(E) ≤ C|E| s © .
Then, the set of Hölder capacities is defined as

(2.2) C(R d ) = ¶ µ ∈ H(R d ) : ∀ E, F ∈ B(R d ), E ⊂ F ⇒ µ(E) ≤ µ(F ) © .
and the set of Hölder Radon measures is defined as

(2.3) M(R d ) = ¶ µ ∈ C(R d ) : µ is a Radon measure © . The topological support supp(µ) of µ ∈ H(R d ) is the set of points x ∈ R d for which µ(B(x, r)) > 0 for every r > 0. A capacity µ is fully supported when supp(µ) = R d .
Similarly, one defines the sets

H([0, 1] d ), C([0, 1] d ) and M([0, 1] d ) by replacing R d by [0, 1] d in the above definitions. Definition 2.2. For s > 0, a set function µ ∈ H(R d ) is s-Hölder when there exists C > 0 such that µ(E) ≤ C|E| s for all E ∈ B(R d ).
Then, for µ ∈ H(R d ), s > 0, and E ∈ R d , define

µ s (E) = µ(E) s and µ (+s) (E) = µ(E)|E| s ,
and if µ is s 0 -Hölder, then for all s ∈ (0, s 0 ), define Starting from µ ∈ H(R d ), µ s and µ (+s) as defined above still belong to H(R d ). If in addition µ is (s + ε)-Hölder, then µ (-s) also belongs to H(R d ).

µ (-s) (E) =        0 if |E| = 0, µ(E)|E| -s if 0 < |E| < +∞, ∞ otherwise. τ µ (t) t 0 -d σµ(H) = τ * µ (H) H 0 d αmin = τ µ (+∞) αmax = τ µ (-∞) τ µ (0)
2.2. Almost-doubling measures and capacities with prescribed multifractal behavior. Multifractal formalisms for measures find their origin in works by physicists who proposed to characterize "strange sets" by considering, for any invariant probability measure µ on such a set S, the partition of S into iso-Hölder sets of µ. They further estimated the "fractal" dimensions of these sets using the Legendre transform of some free energy function, the L q -spectrum, closely related to the Renyi generalized dimensions [START_REF] Hentschel | The infinite number of generalized dimensions of fractals and strange attractors[END_REF][START_REF] Halsey | Fractal measures and their singularities: the characterisation of strange sets[END_REF]. Their ideas were later rigorously formalized by mathematicians (see, e.g. [START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Lau | Multifractal measures and a weak separation condition[END_REF][START_REF] Olsen | A multifractal formalism[END_REF]).

The local behavior of elements of H([0, 1] d ) is described via their pointwise Hölder exponents, also called local dimensions in the case of measures. Definition 2.3. Let µ ∈ H([0, 1] d ). For x ∈ supp(µ), the lower and upper pointwise Hölder exponents of µ at x are respectively defined by

h µ (x) = lim inf j→+∞ log 2 µ(λ j (x))
-j and h µ (x) = lim sup j→∞ log 2 µ(λ j (x)) -j .

Whenever h µ (x) = h µ (x), the common limit is called h µ (x). Then, for α ∈ R,

E µ (α) = ¶ x ∈ supp(µ) : h µ (x) = α © , E µ (α) = ¶ x ∈ supp(µ) : h µ (x) = α © , and
E µ (α) = E µ (α) ∩ E µ (α).
The singularity (or multifractal) spectrum of µ is then the mapping

σ µ : α ∈ R -→ dim E µ (α).
Definition 2.4. The L q -spectrum of µ ∈ H([0, 1] d ) with supp(µ) = ∅ is defined by

τ µ : q ∈ R → lim inf j→+∞ - 1 j log 2 λ∈D j , λ⊂[0,1] d , µ(λ)>0 µ(λ) q .
Then, one always has (see [START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Véhel | Multifractal analysis of Choquet capacities[END_REF])

(2.4) σ µ (α) ≤ τ * µ (α) := inf q∈R qα -τ µ (q).
Definition 2.5. A set function µ ∈ H([0, 1] d ) with supp(µ) = ∅ is said to obey the multifractal formalism (MF) over an interval I ⊂ R when for all α ∈ I,

(2.5) σ µ (α) = τ * µ (α).
It is said to obey the strong multifractal formalism (SMF) over I if for all α ∈ I, in addition to (2.5) one as dim E µ (α) = τ * µ (α). When I = R, one simply says that the MF or the SMF holds for µ.

Remark 2.6. Note that the Hölder exponents are sometimes defined as

h µ (x) = lim inf r→0 + log µ(B(x, r)) log(r) and h µ (x) = lim sup r→0 + log µ(B(x, r)) log(r) , or h µ (x) = lim inf j→∞ log 2 µ(3λ j (x)) -j and h µ (x) = lim sup j→∞ log 2 µ(3λ j (x)) -j .
In this case, µ(λ) is replaced by µ(3λ) in the definition of the L q -spectrum. However, in this paper we mainly consider doubling or almost doubling capacities (see Definition 2.8) for which the previous notions of exponents, level sets, singularity spectrum and L q -spectrum do not depend on whether dyadic cubes or centered balls are considered. [START_REF] Lau | Multifractal measures and a weak separation condition[END_REF][START_REF]Inverse problems in multifractal analysis of measures[END_REF] that τ µ (-∞) exists and is finite if and only if τ µ is finite in a neighborhood of 0 -, and in this case τ µ : R → R is a non-decreasing, concave map with τ µ (1) = 0; moreover, in this case

When µ ∈ M([0, 1] d ) is positive, it is known
dom(τ * µ ) = [τ µ (+∞), τ µ (-∞)] = {α ∈ R : τ * µ (α) ≥ 0} ⊂ [0, ∞).
One also has obviously that τ µ (1) = 0. Due to (2.4), this yields σ µ ≤ Id R . Moreover,

τ * µ (α) = α if and only if α ∈ [τ µ (1 + ), τ µ (1 -)]
. Thus, when µ obeys the MF, there must exist D such that σ µ (D) = D [START_REF] Ngai | A dimension result arising from the l q -spectrum of a measure[END_REF]. Also, if µ has full support in [0, 1] d , then τ µ (0) = -d, and τ * µ reaches its maximum, equal to d, exactly over the interval [τ µ (0) -, τ µ (0) + ].

Definition 2.7. Let T d,M be the set of concave increasing functions τ : R → R such that τ (1) = 0, τ (0) = -d and dom(τ * ) is a compact subset of (0, +∞). Let S d,M be the set of functions σ : R → [0, d] ∪ {-∞} such that σ is compactly supported with support included in (0, +∞), concave, continuous, σ ≤ Id R and there exist two exponents D, D > 0 such that σ(D) = D and σ(D ) = d.

One easily checks that these two sets S d,M and T d,M are Legendre transforms of each other. The set S d,M is the class of admissible singularity spectra for measures strongly obeying the MF with a singularity spectrum compactly supported on (0, ∞), and T d,M is the class of corresponding admissible L q -spectra.

Note that if µ ∈ H([0, 1] d ) has a non-empty support and s > 0 one has σ µ s = σ µ (•/s), and (see the proof of Theorem 2), the set S d defined in Conjecture 1 is related to S d,M by the formula

(2.6) S d = {σ(s•) : σ ∈ S d,M , s > 0}.
Consequenly, in view of the comments made before the statement of Theorem 2, to establish this theorem we are going to construct, for every σ ∈ S d,M , a fully supported µ ∈ M([0, 1] d ) obeying the SMF and satisfying σ µ = σ. The measure we obtain will possess additional properties introduced now.

Definition 2.8. Let Φ be the set of non decreasing functions φ :

N → R + such that lim j→+∞ φ(j) j = 0 . A capacity µ ∈ C(R d ) is almost doubling when there is φ ∈ Φ such that (2.7)
for all x ∈ supp(µ) and j ∈ N, µ(3λ j (x)) ≤ e φ(j) µ(λ j (x)).

Equivalently, there is a mapping φ : (0, 1] → R + such that lim r→0 + φ(r) log(r) = 0 and for all x ∈ supp(µ) and r ∈ (0, 1] one has

µ(B(x, 2r)) ≤ e φ(r) µ(B(x, r)).
When φ is constant, the capacity µ is doubling in the usual sense. Definition 2.9. A set function µ ∈ H(R d ) satisfies property (P) if there exist C, s 1 , s 2 > 0 and φ ∈ Φ such that:

(P 1 ) for all j ∈ N and λ ∈ D j , (2.8) C -1 2 -js 2 ≤ µ(λ) ≤ C2 -js 1 .
(P 2 ) for all j, j ∈ N with j ≥ j, for all λ, λ ∈ D j such that ∂λ ∩ ∂ λ = ∅, and λ ∈ D j such that λ ⊂ λ:

(2.9)

C -1 2 -φ(j) 2 (j -j)s 1 µ(λ ) ≤ µ( λ) ≤ C2 φ(j) 2 (j -j)s 2 µ(λ ).
For µ ∈ H(R d ), (P 1 ) is a uniform Hölder control, from above and below, of µ, and (P 2 ) is a rescaled version of (P 1 ), which implies the almost doubling property. Our result on prescription of multifractal behavior for measures is the following. Theorem 4. There exists a family of measures M d in M(R d ) such that :

(1) Every µ ∈ M d is Z d -invariant, fully supported on R d , satisfies property (P), and

µ |[0,1] d obeys the SMF. (2) S d,M = {σ µ |[0,1] d : µ ∈ M d }.
The family M d ⊂ M(R d ) is built in Section 3, by constructing, for every σ ∈ S d,M , one fully supported Borel probability measure µ on [0, 1] d , which obeys the SMF, and such that σ µ = σ. Then M d is obtained by periodisation of such measures µ.

To prove Theorem 2, and the Frisch-Parisi conjecture in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF], the following class of capacities is needed.

Definition 2.10. The set E d ⊂ C(R d ) is defined as the set of positive powers of measures µ ∈ M d , i.e.
(2.10)

E d = {µ s : µ ∈ M d , s > 0}.
An element of E d is called a multifractal environment.

Theorem 4 and (2.6) have the following immediate consequence.

Corollary 5.

(1) Every µ ∈ E d is Z d -invariant, fully supported on R d , satisfies property (P), and

µ |[0,1] d obeys the SMF. (2) S d = {σ µ |[0,1] d : µ ∈ E d }.
Remark 2.11. As important examples of fully supported Z d -invariant doubling multifractal capacities satisfying property (P) and whose restriction to [0, 1] d obey the multifractal formalism, let us mention the Gibbs capacities obtained as ν s , where s > 0 and ν is a Gibbs measure associated with a Z d -invariant real valued Hölder potential on R d (see [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF]Remark 1.15] for more details). The L q -spectrum of σ ν s |[0,1] d is then analytic. Let us mention an interesting open question: By Theorem 4, each σ ∈ S d is the multifractal spectrum of a capacity satisfying property (P). Which mappings σ ∈ S d can be obtained as singularity spectrum σ µ associated with a doubling capacity µ? 2.3. A function space in which typical functions have a prescribed multifractal spectrum. Let {φ, {ψ (i) } i=1,...,2 d -1 } be a family of wavelets defining a multi-resolution analysis with reconstruction in L 2 (R d ) (see [49, Ch. 2 and 3] for a general construction).

Let Λ = j∈Z Λ j , where for a given j ∈ Z

Λ j = {(i, j, k) : i ∈ {1, . . . , 2 d -1}, k ∈ Z d }.
For every λ = (i, j, k) ∈ Λ, denote by

ψ λ the function x → ψ (i) (2 j x -k).
By the standard results of [START_REF] Meyer | Ondelettes et opérateurs I[END_REF], the two family of functions {2 dj/2 ψ λ } j∈Z,λ∈Λ j and {φ(

• - k)} k∈Z ∪ {2 dj/2 ψ λ } j∈N,λ∈Λ j form two orthonormal bases of L 2 (R d ), and every f ∈ L 2 (R d ) can be expanded as f = k∈Z d β(k)φ(• -k) + j∈N λ∈Λ j c λ ψ λ = j∈Z λ∈Λ j c λ ψ λ , where (2.11) β(k) = R d f (x)φ(x -k) dx and c λ = R d 2 dj ψ λ (x)f (x) dx (k ∈ Z d , λ ∈ Λ).
Pay attention to the L ∞ normalisation used to define the wavelet coefficients (c λ ) λ∈Λ . The two decompositions of f will be used, but the first one is the most important one in the following.

It is known since the seminal works by Jaffard (see [START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF] for a survey) that wavelet coefficients are useful to analyze the pointwise regularity and to understand the multifractal behavior of functions. They are also key in this paper, as well as in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF]. Definition 2.12. For every r ∈ N, call F r the set of those ¶ φ, {ψ (i) } i=1,...,2 d -1 © which define a multi-resolution analysis with reconstruction in L 2 (R d ), and such that φ and the ψ (i) are compactly supported, r times continuously differentiable functions, and every ψ (i) has r vanishing moments, that is for every multi-index α ∈ N d of length smaller than or equal to r, [START_REF] Meyer | Ondelettes et opérateurs I[END_REF]Prop. 4, section 3.7] for instance). Fix r ∈ N * and Ψ ∈ F r , and for any f ∈ L ∞ (R d ), define the sequences (β(k)) k∈Z d and (c λ ) λ∈Λ as in (2.11). Recall that the Hölder-Zygmund spaces {C s (R d )} s>0 are subspaces of L ∞ (R d ), which have the following wavelet characterisation (see [START_REF] Meyer | Ondelettes et opérateurs I[END_REF]Ch. 6] or [START_REF] Triebel | A note on wavelet bases in function spaces[END_REF]): If 0 < s < r, then (2.12)

R d x α 1 1 • • • x α d d ψ (i) (x)dx = 0. It is standard that F r = ∅ for all r ∈ N (see
f ∈ C s (R d ) ⇐⇒      β ∈ ∞ (Z d ), (ε j ) j∈N ∈ ∞ (N), where ε j = Ä 2 js c λ ä λ∈Λ j ∞ (Λ j )
and in this case the decomposition

f = k∈Z d β(k)φ(• -k) + j∈N λ∈Λ j c λ ψ λ holds.
Let us now introduce two functions spaces B µ Ψ (R d ) and B µ Ψ (R d ) associated with any µ ∈ H(R d ) satisfying property (P 1 ) of Definition 2.9 and any enough regular Ψ. Recall Definition 2.2, and observe that as soon as µ ∈ H(R d ) satisfies property (P 1 ), the set functions µ (+s) , and µ (-s) of Definition 2.2 satisfy (P 1 ) as well (when s is small enough in the case of µ (-s) ), and the same is true for (P). Definition 2.13. Write µ(λ) = µ(λ j,k ) for every λ = (i, j, k).

Let µ ∈ H(R d ) satisfy property (P 1 ) of Definition 2.9 with exponents 0 < s 1 ≤ s 2 , and consider an integer n ≥ s 2 + 1.

Fix a wavelet Ψ ∈ F n , and consider for a function f ∈ L p (R d ) the quantity

(2.13) |f | µ,Ψ = (ε µ j ) j∈N ∞ (N) , where ε µ j = Ç c λ µ(λ) å λ∈Λ j ∞ (Λ j ) .
Then, define

B µ Ψ (R d ) = ¶ f ∈ L ∞ (R d ) : |f | µ,Ψ < +∞ © and B µ Ψ (R d ) = 0<ε<min(s 1 ,1) B µ (-ε) Ψ (R d ). (2.14) If B µ Ψ (R d ) (resp. B µ Ψ (R d )) does not depend on Ψ, we simply denote it by B µ (R d ) (resp. B µ (R d )). The space (B µ Ψ (R d ), L ∞ (R d ) + | | µ,Ψ
) is a Banach space, and since B µ (-ε)

Ψ (R d ) → B µ (-ε ) Ψ (R d ) for all 0 < ε < ε < min(1, s 1 ), B µ Ψ (R d
) can be endowed with a Fréchet space structure (Section 5.2 exhibits a basis of neighborhhoods of the origin).

Note that

B µ Ψ (R d ) → C s 1 (R d ), and if µ = (L d ) s/d , then B µ Ψ (R d ) = C s (R d
), so we recover the independence with respect to the choice of Ψ stated in (2.12). In general, we can get such a property for the larger space B µ Ψ (R d ) only. Indeed, the following result is proved in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF], where we investigate a generalisation of the classical Besov spaces, of which Hölder-Zygmund spaces are special cases.

Theorem 6 ([12]

). Assume that µ is an almost-doubling capacity satisfying property (P). The Fréchet space

B µ Ψ (R d ) is independent of the choice of the wavelet Ψ ∈ F n used to define the spaces B µ (-ε) Ψ (R d ). The same holds for B µ Ψ (R d ) if
µ is doubling and property (P) holds for µ with φ equal to 0.

The multifractal properties of functions in B µ (R d ) are now stated when µ ∈ E d (recall that in this case Property (P) holds).

Theorem 7. Let µ ∈ E d be a multifractal environment.

(1) For all f ∈ B µ (R d ),

σ f (H) ≤ σ µ (H) = τ * µ (H) if H ≤ τ µ (0 -) d if H > τ µ (0 -). (2.15) (2) For typical functions f ∈ B µ (R d ), one has σ f = σ µ .
Note that the same holds in B µ Ψ (R d ). Combining the previous results, we can prove Theorem 2.

Proof of Theorem 2. Let σ ∈ S d . Let σ M = σ(•/s), where s is the unique positive real number such that σ(•/s) ≤ Id R and there exists at least one H such that σ(H/s) = H. In other words, s is the unique real number such that σ * (s) = 0. In particular, σ M ∈ S d,M . By Theorem 4, there exists µ ∈ M d such that σ µ = σ M . Moreover, both µ and µ s obey the SMF. Now, we apply Theorems 7 with the capacity µ s ∈ E d : in the Baire space B µ s (R d ), typical functions have σ as singularity spectrum.

Section 3 is dedicated to the construction of the class of measures M d (Definitions 3.8 and 3.14) with prescribed multifractal behavior as described in Theorem 4.

In section 4, the upper bound for the singularity spectrum of all functions in B µ (R d ) is obtained (part (1) of Theorem 7). Part (2) of Theorem 7 is shown in Section 5. It consists first in exhibiting a specific function g µ whose singularity spectrum equals σ µ , and then, by perturbation of the wavelet coefficients of g µ , constructing a dense G δ -set in B µ (R d ) in which all functions share the same multifractal spectrum as g µ .

Finally, in Section 6, some properties relating capacities (and associated auxiliary measures) belonging to E d with the dyadic approximation in R d are proved. These results are not necessary in this paper to obtain our main results, but play a major role in the solution to the Frisch-Parisi conjecture obtained in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF].

Measures with prescribed multifractal behavior

In Section 3.1, additional general properties associated with multifractal formalism for capacities are recalled. Section 2.2 is a preparation to the construction of the measures satisfying the requirements of Theorem 4. The construction is achieved when d = 1 in Section 3.3. Then, in Sections 3.4 to 3.6 the conclusions of Theorem 4 are obtained. The construction is extended to the case d ≥ 2 in Section 3.7 and we get the desired family of measures M d . Finally, in Sections 6.1 and 6.2 we investigate some connections between the elements of E d and metric number theory: a ubiquity theorem associated with µ ∈ E d and the family of dyadic vectors is established, and it is proved that auxiliary measures associated with µ ∈ E d are supported on the set of points which are badly approximated by dyadic vectors. These properties are necessary to achieve the multifractal analysis of typical elements of the Baire spaces considered in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF].

3.1. Additional notions related to the multifractal formalism for capacities. Let us introduce the following notations for ε > 0, α ∈ R, and I = [a, b] an interval:

α ± ε = [α -ε, α + ε] and I ± ε = [a -ε, b + ε]. (3.1)
Also, the convention log(0) = -∞ is adopted. Next propositions complete the properties mentioned in Section 2.2 about multifractal analysis of capacities (some of these properties will be used only in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF]). Recall that the Legendre spectrum α → τ * µ (α) is increasing on the interval α ≤ τ µ (0 -), and is decreasing on α ≥ τ µ (0 + ). The following Propositions 3.1 and 3.3 are easily deduced from any of the following sources [START_REF]Inverse problems in multifractal analysis of measures[END_REF][START_REF] Brown | On the multifractal analysis of measures[END_REF][START_REF] Olsen | A multifractal formalism[END_REF][START_REF] Véhel | Multifractal analysis of Choquet capacities[END_REF][START_REF] Barral | Comparing multifractal formalisms: the neighboring condition[END_REF].

Proposition 3.1. Let µ ∈ C([0, 1] d ) such that supp(µ) = ∅. For α ∈ R, let E ≤ µ (α) = {x ∈ supp(µ) : h µ (x) ≤ α} and E ≥ µ (α) = {x ∈ supp(µ) : h µ (x) ≥ α}.

One has:

(1)

For every α ≤ τ µ (0 -), dim E ≤ µ (α) ≤ τ * µ (α).
(2) For every α ≥ τ µ (0

+ ), dim E ≥ µ (α) ≤ τ * µ (α)
. The distribution of a capacity at small scales can be described through its large deviations spectrum.

Definition 3.2. Let µ ∈ C([0, 1] d ) with supp(µ) = ∅. For I ⊂ R and j ∈ N * define D µ (j, I) = ® λ ⊂ [0, 1] d , λ ∈ D j : log 2 µ(λ) -j ∈ I
´.

The lower and upper large deviations spectra of µ are defined respectively as

σ LD µ : α ∈ R → lim ε→0 lim inf j→∞ log 2 #D µ (j, α ± ε) j and σ LD µ : α ∈ R → lim ε→0 lim sup j→+∞ log 2 #D µ (j, α ± ε) j . Proposition 3.3. Let µ ∈ C([0, 1] d )
with supp(µ) = ∅, such that µ obeys the SMF (Definition 2.5). One has dom(τ * µ ) = {α ∈ R : τ * µ (α) ≥ 0}, and: (1) For every α ∈ R, one has 4) For every η > 0 and every interval I ⊂ dom(τ * µ ), there exists ε 0 > 0 and J 0 ∈ N such that for every ε ∈ (0, ε 0 ) and j ≥ J 0 , for I ∈ {I, I ± ε},

σ µ (α) = dim E µ (α) = dim E µ (α) = dim E µ (α) = σ LD µ (α) = σ LD µ (α) = τ * µ (α). (2) For every α ≤ τ µ (0 -), dim E ≤ µ (α) = τ * µ (α). (3) For every α ≥ τ µ (0 + ), dim E ≥ µ (α) = τ * µ (α). (
log 2 #D µ (j, I) j -sup α∈I τ * µ (α) ≤ η.
(

) If dom(τ * µ ) is compact, then dom(τ * µ ) = [τ µ (+∞), τ µ (-∞)] 5
and there exists a positive decreasing sequence (ε j ) j≥0 tending to 0 when j → +∞, such that for all j ∈ N and λ ∈ D j ,

τ µ (+∞) -ε j ≤ log 2 µ(λ) -j ≤ τ µ (-∞) + ε j .
We now prove Theorem 4 in the case d = 1.

3.2.

A family of probability vectors associated with σ ∈ S 1,M . Fix σ ∈ S 1,M (recall Definition 2.7). In this section, a sequence of probability vectors (p N ) N ≥1 (where

p N ∈ [0, 1] 2 N
),which constitutes the core of the construction of a measure µ satisfying both (P ) and the MF with τ * µ = σ, is defined. For this, write dom(σ) = [α min , α max ]. When α min = 1 = α max , the Lebesgue measure on [0, 1] yields a solution to the inverse problem studied in this Section 3. So it is assumed from now on that α min < α max .

Let us start by introducing two parameters D and D defined as follows:

• if σ(1) = 1, set D = D = 1. • if σ(1) = 1, let 0 < D < 1 < D be such that σ(D) = D and σ(D ) = 1.
The existence of such exponents D and D is justified by the discussion preceding Definition 2.7.

Then, fix an integer N 0 large enough so that for all N ≥ N 0 , setting ε N = 2 log 2 (N )/N , there exists a finite subset A N = {α N,i : i = 1, ..., 2m N } of [α min , α max ] satisfying:

• ε N 0 ≤ α min /4; • m N ≤ 2N (α max -α min ); • D, D ∈ A N ; • for every i ∈ {1, ..., m N -1}, (4N ) -1 < α N,i+1 -α N,i < N -1 ;
• the following inclusions hold:

(3.2) A N ⊂ σ -1 1 N + ε N , 1 ⊂ m N i=1 α N,i - 1 N , α N,i + 1 N ; • for every i ∈ {m N + 1, ..., 2m N }, α N,i = α N,2m N -i+1 ; • if σ(α min ) > 0, then α N,1 = α min .
The continuity of σ is used to get (3.2), and when D = D the above conditions impose that |D -

D | ≥ (4N ) -1 . Denote by i N (resp. i N ) the index in [1, m N ] such that D = α N,i N (resp. D = α N,i N ). Note that i N = i N if and only if D = D = 1.
For each 1

≤ i ≤ m N such that i ∈ {i N , i N }, set (3.3) R N,i = ö 2 N (σ(α N,i )-ε N )-1 ù , which implies that for every i, 1 ≤ R N,i ≤ 2 N -1 N -2 . When D = D , one has i N = i N and one sets R N,i N = 2 N -1 - m N i=1, i =i N R N,i .
When D < D , one has i N < i N and in this cas one sets

(3.4) R N,i N = 2 N σ(α N,i N )-1 = 2 N D-1 and R N,i N = 2 N -1 - m N i=1, i =i N R N,i .
In all cases, by construction

m N i=1, i =i N R N,i ≤ m N 2 N -1 N -2 + 1 {D =D } 2 N D-1 = o(2 N -1 ) as N → ∞,
since the term 1 {D =D } 2 N D-1 appears if and only if D < 1. This also implies that

(3.5) R N,i N = 2 N -1 (1 + o(1)).
Without restriction, we choose N 0 large enough so that

(3.6) for all N ≥ N 0 , m N i=1, i =i N R N,i ≤ 2 N -2 .
Finally, for N ≥ N 0 and m

N < i ≤ 2m N , set R N,i = R N,2m N -i+1 , so that 2m N i=1 R N,i = 2 N .
Definition 3.4. The collection of exponents (β N,i ) 0≤i≤2 N -1 is defined as follows:

(3.7) for all 1 ≤ j ≤ 2m n , β N,i = α N,j if j-1 k=1 R N,k ≤ i < j k=1 R N,k .
In other words, (β N,i ) 0≤i≤2 N -1 is obtained by repeating R N,1 times the value α N,1 , R N,2 times α N,2 , and so on, until repeating R N,2m N times α N,2m N = α N,1 .

Lemma 3.5. Let p N = (p N,i ) 0≤i≤2 N -1 be the probability vector defined by

p N,i = 2 -N β N,i 2 N -1 j=0 2 -N β N,j . One has p N,0 = p N,2 N -1 , and if |i -i | ≤ 1, then (3.8) p N,i p N,i ∈ [2 -1 , 2].
In addition, for N large enough,

(3.9) p N,i 2 N β N,i = 1 + ε N,i , where ε N,i = O(N -1 ) uniformly in 0 ≤ i < 2 N .
Proof. By definition,

p N,i 2 N β N,i = 1 2 m N j=1 2 -N α N,j R N,j
.

In order to estimate p N,i 2 N β N,i uniformly in i, recall that σ ≤ Id R , so that using the definition of R N,i and ε N , one gets

2 N D-1 2 -N D = R N,i N 2 -N α N,i N ≤ m N i=1 2 -N α N,i R N,i ≤ 1≤i =i N ,i =i N ≤m N 2 N (σ(α N,i )-α N,i -ε N ) + R N,i N 2 -N D + 1 D =D R N,i N 2 -N D ≤ m N N -2 + 2 -N D 2 N D-1 + 1 D =D 2 N (1-D ) .
Also, recall that when D = D , D < 1 and D > 1. Consequently, since 2 N D-1 2 -N D = 1/2 + o(1), the previous sequence of inequalities gives (3.9). The fact that (3.8) holds when 0 ≤ i, i ≤ 2 N -1 and |i -i | ≤ 1 follows from the equality p N,i /p N,i = 2 -N (β N,i -β N,i ) and the fact that

|β N,i -β N,i | ≤ N -1 .
Finally, p N,0 = p N,2 N -1 by definition of these parameters.

3.3.

Construction of the measure µ σ associated with σ ∈ S 1,M . A Moran measure µ σ is iteratively constructed as concatenation of pieces of Bernoulli product measures associated with the probability vectors (p N ) N ≥N 0 . The sequence (p N ) N ≥N 0 has been built so that when N → +∞, the singularity spectrum of the Bernoulli product measure associated with p N pointwise converges to σ. Indeed, each p N has been chosen so that, heuristically, there are 2 N σ(α N,i ) weights of order 2 -N α N,i and the α N,i tend to be more or less uniformly distributed in the domain of σ.

Further ingredients are introduced:

• For N ≥ N 0 , an integer N ≥ N 2 is fixed, such that ( N ) N ≥N 0 forms an increasing sequence;

• consider the product space Σ = ∞ N =N 0 {0, • • • , 2 N -1} N ; • for N ≥ N 0 , if g = + N -1
n=N 0 n with 1 ≤ ≤ N , a word of generation (or length) g

(J N 0 , J N 0 +1 , . . . , J N -1 , J N ) ∈ Σ g := N -1 n=N 0 {0, . . . 2 n -1} n × {0, . . . 2 N -1} is also denoted J N 0 • J N 0 +1 • • • J N ; then the cylinder consisting of those elements in Σ with common prefix J N 0 • J N 0 +1 • • • J N is denoted [J N 0 • J N 0 +1 • • • J N ],
and the set of such cylinders of generation g is denoted C g ; • the space Σ is endowed with the σ-field B generated by the cylinders. Definition 3.6. The probability measure ν σ on (Σ, B) is defined as follows. For all N ≥ N 0 , for all 1 ≤ ≤ N , for g = + N -1 n=N 0 n and

[J N 0 • J N 0 +1 • • • J N ] ∈ C g , set ν σ ([J N 0 • J N 0 +1 • • • J N ]) = N -1 n=N 0 n k=1 p n,j n,k k=1 p N,k , (3.10)
where :

• for every n ≥ N 0 , for every i ∈ {1, . . . , n }, j n,i ∈ {0, . . . , 2 n -1},

• for N 0 ≤ n ≤ N -1, J n = j n,1 • • • j n, n ∈ {0, • • • , 2 n -1} n , • J N = j N,1 • • • j N, ∈ {0, • • • 2 N -1} .
Remark 3.7. Formula (3.10) could be written

ν σ ([J N 0 • J N 0 +1 • • • J N ]) = N n=N 0 µ n (J n ),
where µ n is the Bernoulli measure associated with the parameters p n = (p n,i ) i=0,...,2 n -1 .

It is immediate to check that (3.10) is consistent, in the sense that ν σ (Σ) = 1 and for every integers g > g ≥ 1, for every cylinder J ∈ C g , ν σ (J) = J ∈C g ,J ⊂J ν σ (J ).

Using (3.9), one sees that there exists C > 0 such that for each N ≥ N 0 and (J n

) N 0 ≤n≤N ∈ N n=N 0 {0, . . . , 2 n -1} n , ν σ ([J N 0 • J N 0 +1 • • • J N ]) ≤ N n=N 0 Ä (1 + C/n)2 -nα min ä n .
Hence ν σ is atomless since the right-hand side tends to 0 as N tends to infinity.

Every g ∈ N * is decomposed in a unique way under the form g = + N -1 n=N 0 n with N ≥ N 0 and 1 ≤ ≤ N (when N = N 0 , the sum N -1 n=N 0 n is 0). Using this decomposition, one can define the mapping γ :

N * → N * g → γ(g) := N + N -1 n=N 0 n n .
(3.11)

The space Σ provides a natural coding of [0, 1]. Indeed, considering the coding map (3.12)

π : x = Ä (x N,k ) N k=1 ä N ≥N 0 ∈ Σ → ∞ N =N 0 2 - N -1 n=N 0 n n N k=1 x N,k 2 -kN ∈ [0, 1],
for each g ∈ N * , π maps bijectively the elements of C g onto the set of closed dyadic subintervals of generation γ(g) of [0, 1].

Definition 3.8. For every σ ∈ S 1,M , consider the Borel probability measure on [0, 1]

µ σ = ν σ • π -1 ,
where ν σ is the measure constructed above in (3.10). Then, µ σ is defined as the natural periodized version of µ σ , i.e. the Z-invariant measure

µ σ : B ∈ B(R) → k∈Z µ((B ∩ [k, k + 1)) -k).
Finally, set

M 1 = {µ σ : σ ∈ S 1,M } ⊂ M(R).
The measures µ σ and µ σ are said to be associated with σ ∈ S 1,M .

Proposition 3.9. Every µ ∈ M 1 satisfies the property (P) of Definition 2.9. Moreover, if µ is associated with σ ∈ S 1,M , then µ |[0,1] has σ as multifractal spectrum, and it obeys the SMF on R + .

Observe that since ν σ is atomless and π is 1-to-1 outside a countable set of points, for any closed dyadic subinterval λ of [0, 1] of generation n ∈ γ(N * ), one has µ σ (λ) = ν σ ([w]), where [w] is the unique cylinder of generation γ -1 (n) such that π([w]) = λ.

Next sections are devoted to the proofs of the various properties of µ σ , which, in particular, yield Proposition 3.9.

For the rest of this section, σ ∈ S 1,M is fixed, and we simply denote by µ and ν the measures µ σ and ν σ associated with σ.

The measure µ satisfies property (P).

Lemma 3.10. The measure µ is almost doubling.

Proof. Let g ∈ N * and write it under the form g = + N -1 n=N 0 n ∈ N with N ≥ N 0 and 1 ≤ ≤ N .

First, note that if g, hence N , is large enough, the term 1 + ε N,i in (3.9) is greater than 1/2 and smaller than 3/2. Hence, for any 1

≤ i ≤ 2m N , (3.13) 2 -N (αmax+ ε N ) ≤ p N,i ≤ 2 -N (α min -ε N ) ,
where ( ε N ) N ≥1 is a non-increasing sequence (independent of i) converging to 0.

We start by dealing with the dyadic intervals of generation γ(g).

Consider two closed dyadic subintervals λ and λ of [0, 1] of generation γ(g) such that λ is the left neighbor of λ. By construction, λ and λ are the images by π of two cylinders [J] and [ J] in C g such that, denoting by u the longest common prefix of the words J and J, there exist N 1 ≥ N 0 and 0 ≤ j < 2 N 1 -2 such that J = u • j • v and J = u • (j + 1) • v, where:

(1) there is 1

≤ ˜ 1 ≤ N 1 such that u ∈ N 1 -1 N =N 0 {0, • • • , 2 n -1} n • {0, • • • , 2 N 1 -1} ˜ 1 ;
(2) v and ṽ belong to and: (a) either v and v are empty words, (b) or all letters of v are 0 and all letters of v are as large as possible.

{0, • • • , 2 N 1 -1} N 1 -1-˜ 1 • N -1 N =N 1 {0, • • • , 2 n -1} n • {0, • • • , 2 N - 1} ,
From (3.8), (3.10) and the fact that p n,0 = p n,2 n -1 for every n ≥ N 0 , one deduces

(3.14) 2 -1 ≤ µ(λ) µ( λ) ≤ 2.
Consider now two neighboring intervals λ and λ of generation j, where γ(g) < j ≤ γ(g + 1). Let λ and λ be the elements of D γ(g) which contain λ and λ respectively. These intervals are either equal or neighbors. By construction, when N is large enough, one has by (3.8) and (3.10)

(3.15) 2 -N (αmax+ε) ≤ µ(λ) µ(λ ) ≤ 1,
and if j = γ(g + 1), then

(3.16) 2 -N (αmax+ε) ≤ µ(λ) µ(λ ) ≤ 2 -N (α min -ε) ,
where ε ≤ α min /2 for instance (this is due to the choice of N 0 with ε N 0 ≤ α min /4).

Remark 3.11. Observe that by construction, when g gets larger, the ε in (3.16) can be taken very small and converges to zero as g → +∞, again because of the construction of p N . This remark is used in Remark 3.15 below.

The same property as (3.15) holds true for λ and λ , hence

(3.17) µ(λ ) µ( λ ) 2 -(αmax+ε)N ≤ µ(λ) µ( λ) = µ(λ) µ(λ ) µ(λ ) µ( λ ) µ( λ ) µ( λ) ≤ µ(λ ) µ( λ ) 2 (αmax+ε)N . Let (3.18) φ(j) = 0 if 0 ≤ j < N 0 N 0 (1 + (α max + ε))N if γ(g) ≤ j < γ(g + 1).
Note that φ(j)/j ≤ (1 + (α max + ε))N/γ(g) which tends to 0 as j → ∞. This follows from the fact that

(3.19) γ(g) ≥ N -1 n=N 0 n n > > N 2
as N → ∞ since n ≥ n 2 for all n ≥ N 0 . Hence φ ∈ Φ. Applying (3.14) to λ and λ , and using (3.17), there exists J such that for j ≥ J,

(3.20) 2 -φ(j) ≤ µ(λ) µ( λ) ≤ 2 φ(j) ,
Upon adding a constant to φ (to take into account the small generations j ≤ J), one concludes that µ |[0,1] is almost doubling in the sense of Definition 2.8.

To prove that µ is almost doubling on R, it is enough to observe that by symmetry of the coefficients (p n,0 = p n,2 n -1 ), for any g

∈ N µ |[1-2 -γ(g) ,1] (• + 1 -2 -γ(g) ) = µ |[0,2 -γ(g) ] ,
and then to use the periodicity of µ. Lemma 3.12. The measure µ satisfies (P).

Proof. First, consider subintervals of [0, 1].

Let ε > 0. For N ≥ N 0 and g = + N -1 n=N 0 n with 1 ≤ ≤ N , any dyadic interval λ ∈ D j with γ(g) ≤ j < γ(g + 1) satisfies, if N is large enough 2 -(γ(g)+N )(αmax+ε/2) ≤ µ(λ) ≤ 2 -γ(g)(α min -ε/2) , (use (3.13) for instance). By our choice for N and (3.19), for γ(g) ≤ j < γ(g + 1), N/j converges to 0 as j → +∞. Hence, for j large enough

(3.21) 2 -j(αmax+ε) ≤ µ(λ) ≤ 2 -j(α min -ε) .
So, (2.8) is satisfied with s 2 = α max + ε and s 1 = α min -ε, and some constant C > 0. This yields property (P 1 ). Let us move to (P 2 ). Let g, g ∈ N * , j, j ∈ N * with j > j and N ≥ N ≥ N 0 such that:

• g = + N -1 n=N 0 n with 1 ≤ ≤ N , and γ(g) ≤ j < γ(g + 1),

• g = + N -1
n=N 0 n with 1 ≤ ≤ N , and γ(g ) ≤ j < γ(g + 1). Consider two neighboring dyadic intervals λ, λ ∈ D j , and an interval λ ∈ D j such that λ ⊂ λ.

Due to the doubling property of µ applied to λ and λ, we have

(3.22) 2 -φ(j) µ(λ) µ(λ ) ≤ µ( λ) µ(λ ) = µ( λ) µ(λ) µ(λ) µ(λ ) ≤ 2 φ(j) µ(λ) µ(λ ) .
For J ≤ j, denote by λ |J the unique element of D J which contains λ, and for j < J ≤ j denote by λ |J the unique element λ of D J such that λ ⊂ λ ⊂ λ. We have

µ(λ |γ(g)+N ) µ(λ |γ(g ) ) ≤ µ(λ) µ(λ ) ≤ µ(λ |γ(g) ) µ(λ |γ(g )+N )
.

It is easily seen that N + N = o(j) + o(j -j) as j, j → +∞. Consequently, using the multiplicative structure of µ and (3.13) yields a function φ ∈ Φ, as well as a constant C ≥ 1, depending on µ only, such that

C -1 2 -j φ(j) 2 (j -j)(α min -ε) ≤ µ(λ) µ(λ ) ≤ C2 j φ(j) 2 (j -j)(αmax+ε) . (3.23)
Incorporating (3.23) in (3.22) shows that (P 2 ) holds with the same exponents s 1 and s 2 as in (P 1 ).

Finally, the same arguments as those developed at the end of the proof of Lemma 3.10 ensure that the property true on [0, 1] extends to R.

The

L q -spectrum of µ |[0,1] equals σ * . Let τ = σ * . Since σ ∈ S 1,M , one has τ ∈ T 1,M .
For simplification, denote µ |[0,1] by µ. For all j ∈ N, let

D 0 j = {λ ∈ D j : λ ⊂ [0, 1] d }.
Fix t ∈ R and g = + N -1 n=N 0 n with N ≥ N 0 and 1 ≤ ≤ N . Assume that g is so large that (3.15) holds for every j ≥ γ(g).

First, remark that, for the integers j such that γ(g) ≤ j < γ(g + 1), for every λ ∈ D 0 γ(g) , by (3.15) one has

2 (j-γ(g)) 2 -N |t|(αmax+ε) ≤ λ ∈D j ,λ ⊂λ µ(λ ) t µ(λ) t ≤ 2 (j-γ(g)) 2 N |t|(αmax+ε) .
Since N + (j -γ(g)) = o(γ(g)) as g → +∞, one deduces that (3.24)

λ∈D 0 j µ(λ) t = 2 o(γ(g)) λ∈D 0 γ(g) µ(λ) t .
This shows that it is enough to study lim inf g→+∞

1 -γ(g) log 2 I∈D 0 γ(g)
µ(I) t to find the value τ µ (t) (actually, τ µ will be proved to be a limit, not only a liminf).

• Let us start with the lower bound for τ µ (t). The multiplicative structure defining ν and µ using concatenation of pieces of Bernoulli product measures yields (3.25)

λ∈D 0 γ(g) µ(λ) t = N -1 n=N 0 2 n -1 i=0 p t n,i n • 2 N -1 i=0 p t N,i .
For each n ≥ N 0 , using (3.9), one has

(3.26) C -1 n,t 2 -ntβ n,i ≤ p t n,i ≤ 2 -ntβ n,i C n,t
, where C n,t tends to 1 when n → +∞ (and does not depend on i ∈ {0, ..., 2 n -1}). Hence, using (3.7), the definition of the R n,i and the inequality 2R n,i ≤ 2 nσ(α n,i ) which follows from (3.3), one gets

2 n -1 i=0 p t n,i ≤ C n,t 2 n -1 i=0 2 -tnβ n,i ≤ C n,t mn i=0 2R n,i 2 -tnα n,i ≤ C n,t mn i=1 2 n(σ(α n,i )-tα n,i ) ≤ C n,t m n 2 -n inf{tα-σ(α): α∈dom(σ)} = C n,t m n 2 -nτ (t) .
Consequently,

λ∈D 0 γ(g) µ(λ) t ≤ 2 -γ(g)τ (t) • N -1 n=N 0 (C n,t m n ) n • (C N,t m N ) . Since log(m n ) = o(n), one sees that log(m N )+ N -1 n=N 0 n log m n = o(γ(g)).
Combining this with the fact that (C n,t ) n≥N 0 converges to 1 as n tends to infinity, one deduces that

N -1 n=N 0 (C n,t m n ) n • (C N,t m N ) = 2 o(γ(g)) and τ µ (t) = lim inf g→+∞ -1 γ(g) log 2 λ∈D 0 γ(g) µ(λ) t ≥ τ (t).
• Let us now estimate lim sup g→+∞

-1 γ(g) log 2 λ∈D 0 γ(g)
.

Suppose first that σ(τ (t + )) > 0. By construction, one can fix N 0 ≥ N 0 such that for all n ≥ N 0 , there exists an integer 1 ≤ i n,t ≤ m n , such that |α n,in,t -τ (t + )| ≤ 1/n, i n,t = i n and i n,t = i n . The Legendre transform σ = τ * implies then that tτ (t + ) -τ (t) = σ(τ (t + )).

In addition, by continuity of σ, lim n→+∞ η n = 0, where η n = σ(α n,in,t ) -tα n,in,t + τ (t). Bounding from below the sums in (3.25) by the sum only over those integers j such that β n,j = α n,in,t (see (3.7)), and recalling (3.26) and the definition (3.3) of R n,i ,

λ∈D 0 γ(g) µ(λ) t ≥ N 0 -1 n=N 0 2 n -1 i=0 p t n,i n • N -1 n=N 0 C -1 n,t 2 n(σ(α n,i n,t )-εn) 2 -tnα n,i n,t n • C -1 N,t 2 N (σ(α N,i N,t )-ε N ) 2 -tN α N,i N,t .
Recalling now that ε n = 2 log 2 n n , and setting

C t = N 0 -1 n=N 0 2 n -1 i=0 p t n,i n , one gets λ∈D 0 γ(g) µ(λ) t ≥ C t N -1 n=N 0 C -1 n,t 2 n(σ(α n,i n,t )-tα n,i n,t ) 4n 2 n • C -1 N,t 2
N (σ(α N,i N,t )-tα N,i N,t ) 4N 2 = C t 2 -γ(g)τ (t) N -1 n=N 0 C -1 n,t 2 nηn 4n 2 n • C -1 N,t 2 N η N 4N 2 = 2 -γ(g)(τ (t)+o(1))
as g → +∞, where we used that log(C n,t

) + nη n + log(4n 2 ) = o(n) (recall that C n,t → 1 when n → +∞ uniformly in t). The last lines imply that lim sup g→+∞ -1 γ(g) log 2 λ∈D 0 γ(g) µ(λ) t ≤ τ (t).
This equation and the lower bound already obtained for τ µ (t) show that τ µ (t) = τ (t).

It remains us to consider the extremal case σ(τ (t + )) = 0, which may happen only if τ (t + ) ∈ {α min , α max }.

Suppose that τ (t + ) = α min and σ(α min ) = 0. One has 0 = σ(α min ) = τ * (α min ) = t + τ (t + ) -τ (t), so τ (t) = tα min , and t 0 = min{t ∈ R : τ (t) = α min t} < +∞. In addition, t 0 > 0 since τ (0) < 0. Also, for t ∈ [0, t 0 ), σ(τ (t + )) ∈ (0, 1] and we know from the first part of this proof that τ µ (t) = τ (t) on this interval [0, t 0 ). To conclude, it is thus enough to show that this last equality holds over the whole interval [t 0 , +∞) as well.

At first, for all t ≥ t 0 , ε ∈ (0, t 0 ) and n ∈ N, by subadditivity of x ≥ 0 → x t/(t 0 -ε) ,

λ∈D 0 γ(g) µ(λ) t ≤ λ∈D 0 γ(g) µ(λ) t 0 -ε t/(t 0 -ε) , so (3.27) τ µ (t) = lim inf g→∞ - 1 γ(g) log 2 λ∈D 0 γ(g) µ(λ) t ≥ t t 0 -ε τ (t 0 -ε).
On the other hand, consider the interval [0, 2 -γ(g) ] in D γ(g) . Its µ-mass is by construction 2 -γ(g)(α min +o(1)) as g → +∞, so

lim sup g→+∞ - 1 γ(g) log 2 λ∈D 0 γ(g) µ(λ) t ≤ lim sup g→+∞ - 1 γ(g) log 2 2 -tγ(g)(α min +o(1)) = α min t.
Letting ε → 0 in (3.27) and using that α min = τ (t 0 )/t 0 , yelds τ µ (t) = α min t = τ (t).

The case τ (t + ) = α max and σ(α max ) = 0 works similarly by considering t 0 = max{t ∈ R : τ (t) = α max t} ∈ (-∞, 0), and the element of D 0 γ(g) whose µ-mass is minimal, i.e. equal to 2 -γ(g)(αmax+o(1)) .

3.6. The SMF holds for µ with σ µ = σ. The facts that E µ (α) = ∅ for α ∈ [α min , α max ] and dim E µ (α) ≤ σ(α) for α ∈ [α min , α max ], follow from Proposition 3.1 and Section 3.5 where it is proved that τ

* µ = σ (so τ * µ (α) = -∞ if α / ∈ [α min , α max ]).
Further, it follows from the construction and the choice of the weights p n,i that there exist real numbers x at which h µ (x) = α min , and other real numbers x at which h µ (x) = α max . Hence, σ µ (α min ) ≥ 0 and σ µ (α max ) ≥ 0.

In particular, if σ(α min ) = 0 (resp. σ(α max ) = 0), then σ µ (α min ) = 0 (resp. σ µ (α max ) = 0) and the SMF holds at α min (resp. α max ).

Now, fix α ∈ [α min , α max ] such that σ(α) > 0. For each N ≥ N 0 , let (3.28) J N,α = ¶ j ∈ {0, . . . 2 N -1} : j is odd and |β N,j -α| ≤ N -1 © .
Let ε > 0. Recalling the definitions of Section 3.2 we first observe that the exponents β N,j considered in the definition of J N,α correspond to at most nine distinct exponents α N,i (since α N,i -α N,i-1 ≤ (4N ) -1 ). This observation, together with the continuity of σ and the definition of the numbers R N,i imply that for N large enough,

(3.29) 2 N (σ(α)-ε) ≤ #J N,α ≤ 2 N (σ(α)+ε) .
Consider the measure ν α supported on

Σ α = ∞ n=N 0 J n n,α ⊂ Σ
defined by setting, for each N ≥ N 0 , 0 ≤ < N and for every word

J N 0 • J N 0 +1 • • • J N ∈ N -1 n=N 0 {0, • • • 2 n -1} n × {0, • • • 2 N -1} : ν α ([J N 0 • • • J N ]) = (#J N,α ) -N -1 n=N 0 (#J n,α ) -n if [J N 0 • • • J N ] ∩ Σ α = ∅, 0 otherwise.
This last formula is consistent, and the measure ν α is well-defined and atomless.

Proposition 3.13. The measure µ α = ν α • π -1 is defined as the push-forward measure of ν α on the interval [0, 1] (recall (3.12)). This measure is supported by π(Σ α ), and for every x ∈ π(Σ α ), h µ (x) = α and h µα (x) = σ(α).

Proof. For all ω ∈ Σ, denote by [ω |g ] the cylinder of generation g ∈ N which contains ω.

From the definition of J N,α , for every ω ∈ Σ α one has

α -ε ≤ lim inf g→+∞ - 1 γ(g) log Ä µ(π([ω |g ])) ä ≤ lim sup g→+∞ - 1 γ(g) log Ä µ(π([ω |g ])) ä ≤ α + ε.
Since this holds for every choice of ε > 0,

lim g→+∞ - 1 γ(g) log Ä µ(π([ω |g ])) ä = α.
Moreover, lim g→=+∞ γ(g+1) γ(g) = 1 and µ is almost doubling, so π(Σ α ) ⊂ E µ (α).

On the other hand, from (3.29) one deduces that for every

ω ∈ Σ α σ(α) -ε ≤ lim inf g→+∞ -1 γ(g) log Ä µ α (π([ω |g ])) ä ≤ lim sup g→+∞ -1 γ(g) log Ä µ α (π([ω |g ])) ä ≤ σ(α) + ε.
Again, this holds for every choice of ε > 0, hence

lim g→+∞ - 1 γ(g) log Ä µ α (π([ω |g ])) ä = σ(α).
Since lim g→+∞ γ(g+1) γ(g) = 1, the measure µ α , which is supported by π(Σ α ), is exact dimensional with dimension σ(α), so dim(Σ α ) ≥ σ(α). The combination of the last two facts imply that σ µ (α) = dim E µ (α) ≥ σ(α). Since the converse inequality holds true by the multifractal formalism, the proof is complete.

3.7. The case d ≥ 2. If σ ∈ S d,M , then the map σ : α ∈ R → d -1 σ(d • α) belongs to S 1,M
. Let µ σ be the measure associated with σ as built in the previous sections in dimension 1. It is easily checked that the tensor product measure µ = ( µ σ ) ⊗d possesses all the required properties so that σ = σ µ and µ obeys the SMF. In addition, for all α ∈ dom(σ), if ν d -1 α and µ d -1 α are the measures built in Section 3.6 associated with the exponent d -1 α, then the measure µ α := (( µ d -1 α ) ⊗d ) satisfies the same properties as µ α (see Proposition 3.13).

Definition 3.14. Set M d = {µ ⊗d : µ ∈ M 1 }.
Remark 3.15. By construction, for any µ ∈ M d , writing dom(σ µ ) = [α min , α max ], the inequality (3.21) holds, and the estimates made in the previous sections show that for every ε > 0, there exists j ε ∈ N such that for all j ≥ j ≥ j ε , for all λ, λ ∈ D j such that ∂λ ∩ ∂ λ = ∅, and all λ ∈ D j such that λ ⊂ λ, one has

(3.30) µ(λ ) ≤ µ( λ)2 jε 2 -(j -j)(α min -ε) .
Also, from the construction of µ, for all integers j, j ≥ 0 and λ ∈ D j , one has

(3.31) µ(λ • [0, 2 -j ] d ) = µ(λ)2 -φ λ 2 -j α min + φλ (j ) ,
where:

• λ • [0, 2 -j ] d is the concatenation of λ and [0, 2 -j ] d , meaning that λ • [0, 2 -j ] d is the image of [0, 2 -j ] d by the canonical similarity which maps [0, 1] d onto λ, • φ λ ∈ R and φλ ∈ Φ are uniform o(j) in the sense that (3.32) lim j→+∞ sup ® |φ λ | j : λ ∈ D j ´= lim j →+∞ sup    | φλ (j )| j : λ ∈ j∈N D j    = 0.
These properties are key to prove the optimal upper bound for the singularity spectrum of typical functions in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF].

Upper bound for the singularity spectrum in

B µ (R d ) when µ ∈ E d
For λ = (i, j, k) ∈ Λ j , we identify λ with λ j,k ∈ D j , and write µ(λ) for µ(λ j,k ) and λ ⊂ E for λ j,k ⊂ E.

Recall the definition (2.14) of B µ (R d ). To prove Theorem 7(1) and Theorem 7(2), the notion of wavelet leaders is key.

x 0 ∈ R d , h f (x 0 ) < r if and only lim inf j→∞ log L f λ j (x)
log(2 -j ) < r, and in this case

(4.2) h f (x 0 ) = lim inf j→∞ log L f λ j (x)
log(2 -j ) .

Hence, as observed by Jaffard, and rephrased in the language of the present paper, if the support of σ f is bounded and sufficiently smooth wavelets Ψ are used, then the singularity spectrum σ f of f coincides with the singularity spectrum of the capacity

ν ∈ C(R d ) defined by ν(B) = sup ¶ L f λ : λ ∈ D, λ ⊂ B © for all B ∈ B(R d ).
Proof of Theorem 7 [START_REF] Abry | Irregularities and scaling in signal and image processing: Multifractal analysis[END_REF]. Fix some integer r > max(supp(σ µ )) and Ψ ∈ F r . By definition (2.13) and Theorem 6, f ∈ B µ (R d ) implies that for every ε, there exists a constant C f ≥ 0 such that for every j ≥ 0, for every λ ∈ Λ j , |c λ | ≤ C f µ(λ)2 jε . Without loss of generality, we assume that C f = 1. Let us observe that the hierarchal structure of the capacity µ and inequality (2.9) impose that for every dyadic cube λ ∈ Λ j , for every cube λ ∈ Λ j with j ≥ j and λ ⊂ 3λ,

|c λ | ≤ µ(λ )2 j ε ≤ µ(λ)2 φ(j) 2 j ε 2 -(j -j)s 1 .
Let us choose ε < s 1 /2 and j so large that 2 φ(j) ≤ 2 jε . Then

|c λ | ≤ µ(λ)2 2jε 2 -(j -j)(s 1 -ε) ≤ µ(λ)2 2jε .
In particular,

(4.3) L f λ ≤ µ(λ)2 2jε . Thus, due to (4.2), one has h f (x) ≥ h µ (x) for all x ∈ R d . Then, if H ≤ τ µ (0 -), due to Proposition 3.3(1), the fact that E f (H) ∩ [0, 1] d ⊂ E µ |[0,1] d (H) implies that dim(E f (H) ∩ [0, 1] d ) ≤ τ * µ (H) = σ µ (H), and the Z d -invariance of h µ yields σ f (H) = dim E f (H) ≤ τ * µ (H). For H > τ µ (0 -), the desired inequality σ f (H) ≤ d is trivial. 5. Typical singularity spectrum in B µ (R d )
The environment µ ∈ E d and Ψ ∈ F r for some integer r > max(supp(σ µ )) are fixed. Obviously, (2.14)). This function g µ is referred to as a saturation function, since it has the largest possible wavelet coefficients in B µ (R d ).

g µ ∈ B µ Ψ (R d ) ⊂ B µ (R d ) (recall
Also, since µ is a capacity, every wavelet leader of g µ is explicit: for every λ ∈ Λ j , L g λ = max(|c g λ | : λ ⊂ 3λ) = max(µ(λ ) : λ ∈ Λ j , λ ⊂ 3λ). Then, since µ is almost doubling, the following lemma is immediate. Lemma 5.1. For every x ∈ R, one has h gµ (x) = h µ (x) and lim sup j→∞ log L gµ j (x) log 2 -j = h µ (x). In particular, σ gµ = σ µ = τ * µ . We now compute the singularity spectrum of functions whose wavelet coefficients are of same order as those of g µ at an infinite number of generations j ∈ Z. Proposition 5.2. Let f ∈ B µ (R d ) such that for any L ∈ Z d , there exists an increasing sequence of integers (j n ) n∈N , and a positive sequence (ε n ) n∈N converging to 0 such that for all n ≥ 1 and λ = (i,

j n , k) ∈ Λ jn such that λ jn,k ⊂ L + 3[0, 1] d the inequality 2 -jnεn c gµ λ ≤ |c f λ | holds. Then σ f = σ gµ = τ * µ .
Proof. Fix (j n ) n∈N and (ε n ) n∈N as in the statement. Due to the Z d -invariance of µ, it is enough to prove that dim(E f (H) ∩ [0, 1] d ) = τ * µ (H) for all H ∈ R. We already observed in the previous section that any f ∈ B µ (R d ), for all x ∈ [0, 1] d , h f (x) ≥ h µ (x). In addition, by the assumption made in this proposition, for every ε > 0, for every

x ∈ [0, 1] d , (5.2) h µ (x) = lim sup j→+∞ log L g j (x) log 2 -j ≥ lim inf n→+∞ log L g jn (x) log 2 -jn ≥ lim inf n→+∞ log L f jn (x)2 jnε log 2 -jn ≥ h f (x)-ε.
This implies that:

• for all H > τ µ (0 -), E f (H) ∩ [0, 1] d ⊂ E µ |[0,1] d (H), hence dim(E f (H) ∩ [0, 1] d ) ≤ τ * µ (H) by Proposition 3.3(2). • for all H ∈ [α min , α max ], one has E µ |[0,1] d (H) ⊂ E f (H) ∩ [0, 1] d . So dim(E f (H) ∩ [0, 1] d ) ≥ τ *
µ (H). The combination of the last two facts concludes the proof. 5.2. Typical multifractal behavior in B µ (R d ): proof of Theorem 2(2). Recall property (P) (equations (2.8) and (2.9)) satisfied by µ.

Fix Ψ ∈ F r with some r ≥ s 2 + 1. Recall that the family of Banach spaces

B µ (-ε) Ψ (R d ), • L p (R d ) + | • | µ (-ε) ,Ψ 0<ε≤min(1,s 1 ) is such that B µ (-ε) Ψ (R d ) → B µ (-ε ) Ψ (R d ) for all 0 < ε < ε ≤ min(s 1 , 1
). As a consequence, B µ (R d ) can be endowed with a Fréchet space structure, with a countable basis of neighborhoods of the origin given by (5.3)

ß N m = ß f ∈ B µ (R d ) : f L ∞ (R d ) + |f | µ (-1 m ) ,Ψ < 1 m ™™ m∈N, m>max(1,s -1 1 )
.

For every integer m > m 0 = max(1, s -1 1 ) + 1, set

V m = f ∈ B µ (R d ) : ∀j ≥ 0, ∀λ ∈ Λ j , |c f λ | c gµ λ ∈ m -1 {1, . . . , m 2 } . Then let (5.4) G = lim sup m→∞ (V m + N 2 m log(m) ). Each ≥m V , m ≥ m 0 , is dense in B µ (R d ), so G is a dense G δ set.
When f ∈ G, there exists an increasing sequence (j n ) n≥0 such that f ∈ V jn +N 2 jn log(jn) for all n ≥ 0. Fix L ∈ Z d . For every n, for every λ ∈ Λ jn such that λ ⊂ L + 3[0, 1] d , by definition of V jn and N 2 jn log(jn) , one has

|c f λ | ≥ j -1 n c gµ λ -2 -jn log(jn) c gµ λ 2 jn2 -jn log(jn) ≥ c gµ λ 2 -jnεn
, where ε n tends to 0 as n tends to infinity. By Proposition 5.2, this shows that σ f = σ gµ . As a conclusion, the set of those functions f ∈ B µ (R d ) satisfying σ f = σ gµ is Baire generic.

Two additional properties of the capacities µ ∈ E d

In the next subsections, two finer properties satisfied by the capacities µ ∈ E d are proved. They may look a bit disconnected from the rest of the paper and our purpose, but are key to get the full solution to the Frisch-Parisi conjecture in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF].

6.1.

A conditioned ubiquity property associated with the elements of E d .

Let µ ∈ E d . In this section, we measure the size of the set of those points x ∈ R d which are infinitely often close to dyadic vectors 2 -j k ∈ R d such that the order of magnitude of µ(λ j,k ) is 2 -jα min . Definition 6.1. A dyadic vector 2

-j k, j ∈ N, k ∈ Z d , is irreducible when k ∈ Z d \ (2Z) d .
The irreducible representation of a dyadic vector 2 -j k with j ∈ N and k ∈ Z d is the unique irreducible dyadic vector 2 -j k such that 2 -j k = 2 -j k.

If λ = 2 -j (k + [0, 1] d ) ∈ D j , then its associated irreducible cube is λ := 2 -j (k + [0, 1] d ) ∈ D j , where 2 -j k is the irreducible representation of 2 -j k.

Observe that λ is the dyadic cube of generation j located at the "bottom-left" corner of λ. We can write λ = λ • [0, 2 -(j-j) ] d , with the notations defined in Remark 3.15. Definition 6.2. For δ > 1 and j ≥ 1, let (j) δ be the largest integer in γ(N) ∩ [0, j/δ] (recall the definition (3.11) of the mapping γ).

For any positive sequence η = (η j ) j≥1 , let us define the set

X j (δ, η) =        2 -(j) δ k ∈ [0, 1] d :        k ∈ Z d \ 2Z d , µ Ä 2 -(j) δ (k + [0, 1] d ä ≥ 2 -(j) δ (α min +η (j) δ ), µ Ä 2 -(j) δ k + 2 -j [0, 1] d ä ≥ 2 -j(α min +η j )        .
Recall that by construction and (3.21),

µ Ä 2 -(j) δ (k + [0, 1] d ) ä ≤ 2 -(j) δ (α min -ε) µ Ä 2 -(j) δ k + 2 -j [0, 1] d ä ≤ 2 -j(α min -ε) ,
which are complementary to the inequalities used to defined X j (δ, η). Hence, X j (δ, η) contains irreducible dyadic vectors of generation (j) δ whose µ-mass is controlled both at generation (j) δ and at generation j by the exponent α min (note that (j) δ ∼ j/δ). Definition 6.3. For any positive sequence η = (η j ) j≥1 and any increasing sequence of integers (j n ) n≥1 , set

S(δ, η, (j n ) n≥1 ) = N ≥1 n≥N 2 -(jn) δ k∈X jn (δ,η) (2 -(jn) δ k + 2 -jn [0, 1] d ).
An element y ∈ S(δ, η, (j n ) n≥1 ) satisfies |y -2 -(jn) δ k| ≤ 2 -jn ∼ 2 -δ•(jn) δ for infinitely many dyadic vectors of the form 2 -(jn) δ k ∈ X jn (δ, η): we say that y is approximated at rate δ by the elements of the sets X jn (δ, η), n ≥ 1 (around which µ-mass is locally controlled by α min at generations j n and j n (δ)).

Recall that the lower Hausdorff dimension of a Borel probability measure ν on R d is the infimum of the Hausdorff dimension of the Borel sets of positive ν-measure (see [START_REF] Fan | Sur les dimensions de mesures[END_REF] for instance): dim ν = inf{dim E : E ⊂ R d and ν(E) > 0}. Proposition 6.4. Let µ ∈ E d , and suppose that σ µ (α min ) > 0.

There is a positive sequence η = (η j ) j≥1 converging to 0 when j → +∞ such that for any δ > 1, for any increasing sequence of integers (j n ) n≥1 , there exists a Borel probability measure ν on R d of lower Hausdorff dimension larger than or equal to σ µ (α min )/δ, and such that ν(S(δ, η, (j n ) n≥1 )) = 1.

In particular, dim S(δ, η, (j n ) n≥1 )) ≥ σ µ (α min )/δ.

Note that when µ = (L) s for some s > 0, Proposition 6.4 is already obtained in [START_REF] Jaffard | On the Frisch-Parisi conjecture[END_REF].

Proof. We treat the case d = 1; the case d ≥ 2 follows from an easy adaptation. Without loss of generality, assume that µ ∈ M 1 instead of µ ∈ E 1 : this is obviously equivalent up to some positive power of µ. For simplicity, σ µ is denoted by σ.

Preliminary observation. Recall the construction of µ and Section 3.2. Definition 6.5. Let g = + N -1 n=N 0 n ∈ N * , with N ≥ N 0 and 1 ≤ ≤ N . A real number x ∈ [0, 1] satisfies property P (α min , g) when there exists a word w ∈ Σ g such that x ∈ π([w]) and writing w

= J N 0 •J N 0 +1 • • • J N -1 •J with J n = j n,1 • • • j n, n ∈ {0, ..., 2 n -1} n for n ∈ {N 0 , . . . , N -1} and J N = j N,1 • • • j N, ∈ {0, ..., 2 N -1}
, then all the j n,i are such that β n,j n,i = α min .

It is direct to see that there exists a sequence (η j ) j≥1 such that for all x ∈ [0, 1], for all g ≥ 1, if x satisfies property P (α min , g), then for all 1 ≤ j ≤ γ(g), one has µ(λ j (x)) ≥ 2 -j(α min +η j ) .

Fix such a sequence η = (η j ) j≥1 .

Fix δ > 1 and an increasing sequence of integers (j n ) n≥1 . We are going to construct a Cantor subset K included in S(δ, η, (j n ) n≥1 ) and a Borel probability measure ν supported on K such that for all closed dyadic subcubes λ of [0, 1] d of generation j ≥ 0, one has ν(λ) ≤ 2 -j(δ -1 σ(α min )-ψ(j)) , where the function ψ : N → (0, +∞) tends to 0 as n → ∞. The mass distribution principle (see [START_REF] Falconer | Fractal Geometry[END_REF]) allows then to conclude that dim S(δ, η, (j n ) n≥1 )) ≥ σ(α min )/δ. We proceed in three steps. Notations and definitions of Section 3.6 are adopted.

Step 1: Construction of a family of measures (ν λ ) λ∈D .

A family of auxiliary measures indexed by the closed dyadic subintervals of [0, 1] is built in a very similar way as µ α min in Section 3.6.

Let us introduce a notation: for j ∈ N * , set

N (j) = N 0 if 1 ≤ j ≤ N 0 N 0 , N if j > N 0 N 0 and γ( N -1 n=N 0 n ) < j ≤ γ( N n=N 0 n ). Observe that (6.1) lim j→+∞ N (j) j = 0.
Let N ≥ N 0 + 1, 1 ≤ ≤ N , and g = + N -1 n=N 0 n . Let J be an integer such that γ(g -1) < J ≤ γ(g). Note that J ≥ j 0 := N 0 N 0 + 1.

Fix λ ∈ D J , and construct a measure ν λ supported on λ as follows.

For each n ≥ N = N (J), consider (6.2) J n,α min = {j ∈ {0, . . . 2 n -1} : j is odd and β n,j = α min }.

Using (3.3) and (3.4), one sees that for an n ≥ N , (6.3) #J n,α min ≥ 2 n(σ(α min )-2εn) .

Writing λ = K2 -J + 2 -J [0, 1], denote by λ g ⊂ λ the dyadic subinterval K2 -J + 2 -γ(g) [0, 1] and [w λg ] the unique cylinder such that π([w λg ]) = λ g . Observe that [w λg ] ∈ C g , the set of cylinders of generation g in Σ. Then, consider the set

Σ λ = {w λg } × (J N,α min ) N -× n=N +1 (J n,α min ) n ⊂ Σ,
and for each n ≥ N and w ∈ Σ g × {0, . . . ,

2 N -1} N -× n k=N +1 {0, . . . , 2 k -1} k set ρ λ ([w]) = (#J N,α min ) -N + n k=N +1 (#J k,α min ) -k if [w] ∩ Σ λ = ∅ 0 otherwise.
This yields an atomless measure ρ λ whose support is Σ λ . Finally, the measure

ν λ = ρ λ •π -1 is a probability measure supported on λ g ⊂ [0, 1].
By construction of ν λ , using (6.3), for g ≥ g and λ ∈ D γ(g ) , one has either ν λ (λ ) = 0, or λ ∩ π(Σ λ ) = ∅ and

ν λ (λ ) ≤ 2 -(γ(g )-γ(g))(σ(α min )-2ε N (J) ) ≤ 2 -(γ(g )-J)(σ(α min )-2ε N (J) ) 2 N (J)σ(α min ) .
Consequently, for every g ≥ g and γ(g ) < j ≤ γ(g + 1), for λ ∈ D j one has (6.4) ν λ (λ ) ≤ 2 -(j-J)(σ(α min )-2ε N (J) ) 2 2N (j)σ(α min ) .

This inequality extends easily to all integers j such that J ≤ j ≤ γ(g) and λ ∈ D j .

Remark 6.6. By construction, since only odd integers j are considered in the definition of the sets J n,α min , if λ λ and ν λ ( λ) > 0, then λ = λ j, k with k2 -j irreducible. Moreover, writing γ( g) < j ≤ γ( g + 1), if property P (α min , g) of Definition 6.5 holds for all x ∈ λ, then P (α min , g) holds for all x ∈ λ.

We finally set ν λ = ν [0,2 -j 0 ] if λ ∈ j 0 -1 j=1 D j and λ ⊂ [0, 1].

Step 2: Construction of a Cantor set K ⊂ S(δ, (η j ) j≥1 , (j n ) n≥1 ) and a Borel probability measure ν supported on K.

Recall that j 0 = N 0 N 0 + 1. Define n 1 = 0, G 1 = {[0, 2 -j 0 ]} and a set function ν on G 1 by ν([0, 2 -j 0 ]) = 1. Note that γ( N 0 ) < j 0 ≤ γ( N 0 + 1), and that for all x ∈ [0, 2 -j 0 ], property P (α min , N 0 ) holds (recall Definition 6.5).

Let p be a positive integer. Suppose that p families G 1 , . . . , G p of closed dyadic intervals, as well as p integers 0 = n 1 < n 2 < • • • < n p are constructed such that: Let us explain how to build n p+1 and G p+1 . Write γ(g k ) < j np ≤ γ(g k + 1), where g k = + N -1 n=N 0 n ∈ N with N ≥ N 0 and 1 ≤ ≤ N . Fix n p+1 so that γ(g k + 1) ≤ (j n p+1 ) δ (other constraints on n p+1 will be given a few lines below).

Consider λ ↑ ∈ G p . For every λ ∈ D (jn p+1 ) δ with λ ⊂ λ ↑ and ν λ ↑ ( λ) > 0, (6.4) gives ν(λ ↑ )ν λ ↑ ( λ) ≤ ν(λ ↑ )2 -((jn p+1 ) δ -jn p )(σ(α min )-2ε N (jn p ) ) 2 2N ((jn p+1 ) δ )σ(α min ) .

By (6.5) applied to ν(λ ↑ ), and then (6.1), choosing n p+1 large enough yields that ν(λ ↑ )ν λ ↑ ( λ) ≤

2

-jn p+1 (δ -1 σ(α min )-3ε N (jn p ) ) (the equivalence (j n p+1 ) δ ∼ j n p+1 /δ was used).

Further, one sets (6.6)

G p+1 = λ ↑ ∈Gp k2 -(jn p+1 ) δ + 2 -jn p+1 [0, 1] : λ = k2 -(jn p+1 ) δ + 2 -(jn p+1 ) δ [0, 1] ⊂ λ ↑ ν λ ↑ ( λ) > 0 .
By construction, G p+1 ⊂ D jn p+1 , and each interval λ ∈ G p+1 is the left-most interval inside the corresponding interval λ ∈ D (jn p+1 ) δ . It follows from this, (c) and Remark 6.6 that property (c) holds at generation p + 1 as well.

Next, for every λ ∈ G p+1 associated with λ ∈ D (jn p+1 ) δ and λ ↑ ∈ G p , one finally sets ν(λ) = ν(λ ↑ )ν λ ↑ ( λ).

The previous construction and the above remarks show that all the items (a)-(g) above hold with p + 1 as well. Finally, we define K = p≥1 λ∈Gp λ, and the set function ν defined on the elements of p≥1 G p extends to a Borel probability measure on [0, 1], whose topological support is K. It is direct to check that ν is atomless, and that due to property (d) and the preliminary observation, K ⊂ S(δ, η, (j n ) n≥1 ).

Step 3: Let us study the Hölder properties of ν to get a lower bound for its lower Hausdorff dimension.

Fix a closed dyadic subinterval λ in [0, 1] of generation j ≥ j n 2 such that the interior of λ intersects K. Let p ≥ 2 be the smallest integer such that the interior of λ intersects at least two elements of G p . Necessarily, j ≤ j np .

Let λ ↑ be the unique element of G p-1 such that the interior of λ intersects λ ↑ . Since ν is atomless, ν(λ) ≤ ν(λ ↑ ). In addition, ν(λ) = ν(λ ↑ )ν λ ↑ ( λ) where λ is associated with λ as in (6.6). Consequently, for every p, denoting ε N (jn p ) simply by p , if j ≤ j n p-1 then ν(λ) ≤ ν(λ ↑ ) ≤ 2 -jn p-1 (δ -1 σ(α min )-3 p-2 ) ≤ 2 -j(δ -1 σ(α min )-3 p-2 ) , and if j > j n p-1 , then by (6.4) and (6.5), one has ν(λ) = ν(λ ↑ )ν λ ↑ ( λ) ≤ 2 -jn p-1 (δ -1 σ(α min )-3 p-2 ) 2 -(j-jn p-1 )(σ(α min )-2 p-1 ) 2 2N (j)σ(α min ) = 2 -j(δ -1 σ(α min )-ϕ(λ)) , where ϕ(λ) = 3 p-2 + (j -j n p-1 )(σ(α min )(δ -1 -1) + 3 p-2 -2 p-1 ) + 2N (j)σ(α min ) j .

Pay attention to the fact that in the formula above, p depends a priori on λ and j. However, this dependence can be uniformly controlled. Indeed, observe that ϕ(λ) ≤ 6 p-2 + 2N (j)σ(α min ) j , and that when j tends to +∞, min{p ≥ 2 : ∃ λ ∈ D j such that the interior of λ intersects at least 2 elements of G p } also tends to +∞. Consequently, p-2 converges uniformly to 0 over {λ ∈ D j , Int(λ)∩K = ∅} as j → +∞. Thus, remembering that (6.1) holds as well, one concludes that there exists a function ψ : N → (0, +∞) such that lim j→+∞ ψ(j) = 0 and for every λ ∈ D 0 j , ν(λ) ≤ 2 -j(δ -1 σ(α min )-ψ(j)) . In particular the lower Hausdorff dimension of ν is greater than σ(α min )/δ. Since K ⊂ S(δ, η, (j n ) n≥1 ), ν(K) = 1, we get dim S(δ, η, (j n ) n≥1 ) ≥ δ -1 σ(α min ), and the conclusions of Proposition 6.4 holds in dimension 1.

6.2. The set of badly approximated points supports the auxiliary measures µ α . The measures µ α described in Proposition 3.13 are supported on the set of points which are badly approximated by dyadic vectors, as stated by the following lemma. This property is key for the study of typical singularity spectra in [START_REF] Barral | The Frisch-Parisi conjecture II: Besov spaces in multifractal environnement, and a full solution[END_REF]. Proposition 6.7. Let µ ∈ E d . For every x ∈ [0, 1] d , call λ j (x) ∈ D j(x) the irreducible representation of λ j (x). For every α ∈ [α min , α max ] such that τ * µ (α) > 0, for µ α -almost every x, one has lim n→+∞ j n(x) jn = 1.

Proof. Fix α ∈ [α min , α max ] and δ > 1. For j ∈ N * , let E µ (α, δ, j) = {x ∈ E µ (α) : j(x) j ≤ δ -1 } and E µ (α, δ) := x ∈ E µ (α) : lim inf j→+∞ j(x) j ≤ δ -1 = lim sup j→+∞ E µ (α, δ, j).

Figure 1 .

 1 Figure 1. Estimated singularity spectrum (right) for the 1D velocity of a turbulent flow (left) -Credit to P. Abry, H. Wendt

Figure 2 .

 2 Figure 2. Typical multifractal spectrum of probability measures (left) or functions in B s,p q (R d ) when s > d/p (right).

Figure 3 .

 3 Figure 3. Left: Free energy function of µ ∈ C([0, 1] d ) satisfying the MF. Right: The singularity spectrum of µ.

Definition 4 . 1 (

 41 Wavelet leaders). Given Ψ ∈ r∈N F r and f ∈ L p loc (R d ) for p ∈ [1, +∞],denoting the wavelet coefficients of f associates with Ψ by (c λ ) λ∈Λ , the wavelet leader of f associated with λ ∈ D (see Section 2.1 for the notations) is defined as:

  λ | : λ = (i, j, k) ∈ Λ, λ j,k ⊂ 3λ}.Pointwise Hölder exponents of Hölder continuous functions (recall (1.1)) are related to the wavelet leaders as follows (see[START_REF] Jaffard | Wavelet techniques in multifractal analysis[END_REF] Corollary 1]).

Proposition 4 . 2 .

 42 Let r ∈ N * and Ψ ∈ F r . If f ∈ C ε (R d )for some ε > 0, then for every

5. 1 .

 1 A saturation function. Let us set (5.1) g µ (x) = j≥0 λ∈Λ j c gµ λ ψ λ (x), where c gµ λ = µ(λ).

  (a) for every k ∈ {2, ..., p}, (j n k ) δ ≥ j 0 ; (b) for every k ∈ {2, ..., p},G k ⊂ {x + 2 -jn k [0, 1] d : x ∈ X jn k (δ, η)} ⊂ D jn k ; (c) for every k ∈ {1, ..., p}, writing γ(g k ) < j n k ≤ γ(g k + 1) for some integer g k , everyx ∈ G k satisfies property P (α min , g k ); (d) for every k ∈ {2, ..., p}, the irreducible intervals {λ : λ ∈ G k } are pairwise disjoint; (e) for every k ∈ {2, ..., p} and every element of λ ∈ G k , there is a unique λ ↑ ∈ G k-1 such that λ ⊂ λ ⊂ λ ↑ ; (f) the measure ν is defined on the σ-algebra generated by the elements of p k=1 G k by the following formula: for all 2 ≤ k ≤ p and λ ∈ G k ,ν(λ) := ν(λ ↑ )ν λ ↑ (λ); (g) for all 2 ≤ k ≤ p and λ ∈ G k , (6.5) ν(λ) ≤ 2 -jn k Ä δ -1 σ(α min )-3ε N (jn k-1 )ä .

For ε > 0, let F µ (α, j, ε) = {x ∈ [0, 1] d : ∀ j ≥ j, 2 -j (α+ε) ≤ µ(λ j (x)) ≤ 2 -j (α-ε) }.

Setting j δ = j/δ , the following inclusion holds :

Using Proposition 3.3(1) or ( 4), for every fixed ε > 0, one sees that the cardinality of

when j is large. Combining this with the previous embedding, coverings of E µ (α, δ) are obtained using sets of the form j≥J λ j δ ,k ∈D j δ :

), and it is easily seen that dim E µ (α, δ) ≤ τ * µ (α)/δ. This implies that µ α (E µ (α, δ)) = 0, again because µ α may give a positive mass to a set E only when dim E ≥ τ * µ (α). Since this holds for all δ > 1, lim inf j→∞ j(x) j = 1 for µ α -almost every x, and in particular lim n→∞ jn(x) jn = 1.