The multifractal nature of heterogeneous sums of Dirac masses

Abstract

This article investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masseswhere (x n ) n≥0 is a sequence of points in [0,1] d and (w n ) n≥0 is a positive sequence of weights such that n≥0 w n < ∞. We consider the case where the points x n are roughly uniformly distributed in [0,1] d , and the weights w n depend on a random self-similar measure µ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λ n ) n≥1 converging to 0 in the following wayThe measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(x n , λ n )} n with respect to µ.

Threshold and Hausdorff Spectrum of Discontinuous Measures

“…This linear increasing part in the large deviations spectrum is conform to the observations made on special classes of homogeneous and heterogeneous sums of Dirac masses studied in the last fifteen years [1,15,9,23,2,4]. Moreover, the elements of these classes of measures, to which belong the measures (6) and (11), verify that H g (χ) = H τ (χ) even when q τ (χ) = 1.…”

“…Nevertheless, these measures may have very interesting multifractal spectra [15,1,9,14,6,24,2,4], and there is a need for other relevant parameters. The study of the pair (q τ (χ), H τ (χ)) and their relationships with the so-called large deviations spectrum are achieved in [5] and recalled below in Section 2.…”

“…It turns out that this class of measures has a fruitful structure [4,2]. Theorem 1.1 Let µ be a Gibbs measure as defined in Section 3.2.…”

The singularity spectrum of the inverse of cookie-cutters

“…Other examples of infinite homogeneous [1,17,14] or heterogeneous [3,5] sums of Dirac masses have been studied. Note that the measure ν does not belong to the class of discrete measures considered in these papers, except when µ ϕ is the measure of maximal entropy associated with a system where the mappings g 0 and g 1 are affine maps with same contraction ratio.…”

“…Note that the measure ν does not belong to the class of discrete measures considered in these papers, except when µ ϕ is the measure of maximal entropy associated with a system where the mappings g 0 and g 1 are affine maps with same contraction ratio. Moreover, compared to the measures appearing in [1,17,14,3,5], an additional important property is that ν is naturally associated with a dynamical system.…”