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Constructing self-similar subsets within the fractal support of Lacunary Wavelet Series for their multifractal analysis

Céline Esser, Béatrice Vedel

TL;DR

This work addresses the problem of determining the increasing multifractal spectrum of lacunary wavelet series supported on fractal sets $\mathcal{I}$ where the Hausdorff and upper-box dimensions coincide, and classical Mass Transference Principles may not apply. It introduces quasi-Cantor subsets $\mathcal{K}_{\varepsilon}(b) \subset \mathcal{I}$ with controlled self-similarity across scales to obtain lower bounds for limsup-based dimension questions, and then applies these to lacunary wavelet series with parameters $(\alpha,\eta)$, proving that almost surely $\dim_{\mathcal{H}}\{x: h_f(x) \le h\} = (\eta/\alpha) h$ for $h$ in $[\alpha, \alpha H/\eta]$ (where $H = \dim_{\mathcal{H}}(\mathcal{I})$). The key contributions are (i) a constructive method to bypass MTP limitations via $(\varepsilon,b)$-quasi-Cantor sets, (ii) a transference-type result for limsup sets with centers chosen by Bernoulli laws, and (iii) an extension of Jaffard’s lacunary wavelet spectrum to fractal supports, enabling multifractal analysis on irregular domains. These results provide a principled way to estimate the multifractal structure of irregular signals on fractal supports and offer avenues for future generalizations to multivariate and other random processes on fractals.

Abstract

Given a fractal $\mathcal{I}$ whose Hausdorff dimension matches with the upper-box dimension, we propose a new method which consists in selecting inside $\mathcal{I}$ some subsets (called quasi-Cantor sets) of almost same dimension and with controled properties of self-similarties at prescribed scales. It allows us to estimate below the Hausdorff dimension $\mathcal{I}$ intersected to limsup sets of contracted balls selected according a Bernoulli law, in contexts where classical Mass Transference Principles cannot be applied. We apply this result to the computation of the increasing multifractal spectrum of lacunary wavelet series supported on $\mathcal{I}$.

Constructing self-similar subsets within the fractal support of Lacunary Wavelet Series for their multifractal analysis

TL;DR

This work addresses the problem of determining the increasing multifractal spectrum of lacunary wavelet series supported on fractal sets where the Hausdorff and upper-box dimensions coincide, and classical Mass Transference Principles may not apply. It introduces quasi-Cantor subsets with controlled self-similarity across scales to obtain lower bounds for limsup-based dimension questions, and then applies these to lacunary wavelet series with parameters , proving that almost surely for in (where ). The key contributions are (i) a constructive method to bypass MTP limitations via -quasi-Cantor sets, (ii) a transference-type result for limsup sets with centers chosen by Bernoulli laws, and (iii) an extension of Jaffard’s lacunary wavelet spectrum to fractal supports, enabling multifractal analysis on irregular domains. These results provide a principled way to estimate the multifractal structure of irregular signals on fractal supports and offer avenues for future generalizations to multivariate and other random processes on fractals.

Abstract

Given a fractal whose Hausdorff dimension matches with the upper-box dimension, we propose a new method which consists in selecting inside some subsets (called quasi-Cantor sets) of almost same dimension and with controled properties of self-similarties at prescribed scales. It allows us to estimate below the Hausdorff dimension intersected to limsup sets of contracted balls selected according a Bernoulli law, in contexts where classical Mass Transference Principles cannot be applied. We apply this result to the computation of the increasing multifractal spectrum of lacunary wavelet series supported on .

Paper Structure

This paper contains 9 sections, 21 theorems, 140 equations, 1 figure.

Table of Contents

  1. Introduction and statement of the main results : alternative
  2. Basic definitions and classical results on multifractal analysis
  3. Fractal dimensions
  4. Multifractal analysis and computation via wavelets
  5. Sets with good rate of duplication
  6. Quasi-Cantor subsets of $\mathcal{I}$
  7. LWS on the compact fractal set $\mathcal{I}$
  8. Conclusion and comments
  9. Proofs of the results of Section \ref{['sec:goodrate']}

Key Result

Theorem 1.1

Beresn:06 Let $X$ be a compact set in $\mathbb{R}^d$ and assume that there exist $s\in [0, d]$ and $a,b,r_0>0$ such that for any ball $B$ of center $x \in X$ and of radius $r \leq r_0$. Let $\delta >0$. Given a ball $B = B(x,r)$ with center in $X$, we set $B^{\delta} = B \left(x, r^{\delta} \right).$ Assume that $(B_n)_{n \in \mathbb{N}}$ is a sequence of balls with center in $X$ and radius $r_n$

Figures (1)

  • Figure 1: The regularity of a LWS on $\mathcal{I}$ can be larger than the regularity attained on a subset of the dimension of $\mathcal{I}$.

Theorems & Definitions (29)

  • Theorem 1.1: Mass Transference Principle
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 2.1: Hausdorff measure and Hausdorff and upper-box dimensions
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 19 more