Heterogeneous ubiquitous systems in ℝd and Hausdorff dimension

Abstract

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.

The singularity spectrum of the inverse of cookie-cutters

“…This is a crucial point in the following, since such limsup sets arise naturally when performing the multifractal analysis of the inverse measure ν of µ ϕ . Proposition 3.6 has been obtained in [4], Theorem 2.2 (case ρ = 1).…”

“…The multifractal analysis of the discrete measure ν is explained by a subtle combination between the fine local structure of Gibbs measures and the distribution of the jump points of ν. The results rely on the notion of heterogeneous ubiquity introduced in [2,4].…”

“…The notion of heterogeneous ubiquity, introduced in [2,3,4], has been developed to be able to estimate the Hausdorff dimension of limsup sets like the S(ξ, ε). This is a crucial point in the following, since such limsup sets arise naturally when performing the multifractal analysis of the inverse measure ν of µ ϕ .…”

“…In order to apply Theorem 2.2 of [4], three properties are required on the measure µ q0 and the system {(x w , λ w )} w∈Σ * :…”

“…Then, according to the definitions of [2,4], the system of points {(x w , λ w )} w∈Σ * forms an heterogenous ubiquitous system with respect to µ q0 and its almost sure Hölder exponent θ * (θ (q 0 )).…”