Let {xn} n∈N be a sequence of [0, 1] d , {λn} n∈N a sequence of positive real numbers converging to 0, and δ > 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the form S(δ) = N∈N n≥N B(xn, λ δ n ). Let µ be a positive Borel measure on [0, 1] d , ρ ∈ (0, 1] and α > 0. Consider the finer limsup-set Sµ(ρ, δ, α) = N∈N n≥N: µ(B(xn ,λ ρ n ))∼λ ρα n B(xn, λ δ n ).We show that, under suitable assumptions on the measure µ, the Hausdorff dimension of the sets Sµ(ρ, δ, α) can be computed. Moreover, when ρ < 1, a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension of Sµ(ρ, δ, α). Our results apply to several classes of multifractal measures, and S(δ) corresponds to the special case where µ is a monofractal measure like the Lebesgue measure.The computation of the dimensions of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Diophantine approximation conditioned by (or combined with) b-adic expansion properties, by averages of some Birkhoff sums and branching random walks, as well as by asymptotic behavior of random covering numbers.

ABSTRACT. In this work, we study the new class of multifractal measures, which combines additive and multiplicative chaos, defined bywhere µ is any positive Borel measure on [0, 1] and b is an integer ≥ 2.The singularities analysis of the measures νγ,σ involves new results on the mass distribution of µ when µ describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on µ, the multifractal spectrum of νγ,σ is linear on [0, hγ,σ] for some critical value hγ,σ, and then it is strictly concave on the right of hγ,σ, and deduced from the one of µ by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. These measures open interesting perspectives in modeling discontinuous phenomena.

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ABSTRACT. Given a positive locally finite Borel measure µ on R, a natural way to construct multifractal wavelet series

We construct a non-decreasing pure jump Markov process, whose jump measure heavily depends on the values taken by the process. We determine the singularity spectrum of this process, which turns out to be random and to depend locally on the values taken by the process. The result relies on fine properties of the distribution of Poisson point processes and on ubiquity theorems.The corresponding notion of "multifractal function" was put into light by Frisch and Parisi [13], who introduced the definition of the spectrum of singularities of a function f .By monotonicity (if A ⊂ B, then D f (A, h) ≤ D f (B, h)), the limit exists and it is independent of the particular basis chosen. Clearly, a function has a homogeneous spectrum if and only if for all h ≥ 0, D f (t, h) is independent of t ≥ 0. The local spectrum allows one to recover the spectrum of all possible restrictions of f on an open interval.Lemma 4. Let f : R + → R be a locally bounded function. Then for any open interval I = (a, b) ⊂ R + , for any h ≥ 0, we have D f (I, h) = sup t∈I D f (t, h).Proof. Let thus h ≥ 0 be fixed. First, it is obvious that for any t ∈ I, D f (t, h) ≤ D f (I, h), since for (V n ) n≥1 a basis of neighborhoods of t, V n ⊂ I for n large enough. Next, set δ = D f (I, h), and consider ε > 0. We want to find t ∈ I such that for all neighborhood V