Renewal of singularity sets of random self-similar measures

Abstract

This paper investigates new properties concerning the multifractal structure of a class of random self-similar measures. These measures include the well-known

The multifractal nature of heterogeneous sums of Dirac masses

“…The second one contains limits of [0,1] d -martingales considered in the multiplicative chaos in the meaning of [21]. It is shown in [9,10,11] that these measures µ obey conditions P1-3 for suitable systems {(x n , λ n )} n (including those of Section 5·1) when h ranges in the interval where τ • (Random) Gibbs measures.…”

“…The first examples are the independent multiplicative cascades, or Mandelbrot martingales introduced in [27] and then studied extensively in [27,23,18,29,28,1,15,4,5,10]. They are a particular case of a wider class of [0,1] d -martingales -see [7] which satisfy condition P2(µ, ρ, {(x n , λ n )}, h).…”

From Multifractal Measures to Multifractal Wavelet Series

“…We made this dyadic choice for sake of simplicity. Nevertheless, we mention that Theorem 4 also holds for measures that depend on a b-adic grid with b greater than 2 (see [12,13]). …”

“…In this work, we detail the example of dyadic Mandelbrot random multiplicative cascades, and we refer the reader to [12,13] for more details on statistical self-similar measures and for the proof of Theorem 4 in these cases. The wavelet series F µ associated with Mandelbrot cascades is particularly interesting, since the perturbations of such series allows us to derive the Hausdorff multifractal spectrum of the "random wavelet cascades" of Arnéodo, Bacry and Muzy [3].…”

“…We detail hereafter the cases of dyadic random multiplicative cascades and of stable Lévy measures. The properties required to deal with the examples that we mentioned above, but that we do not treat in this paper, can be found in [12,13].…”