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

TL;DR

This paper introduces a new method for constructing self-similar subsets within fractal supports to analyze the multifractal spectrum of lacunary wavelet series, especially where classical principles fail.

Contribution

It develops a novel approach to estimate Hausdorff dimensions of intersections with limsup sets, enabling multifractal analysis of lacunary wavelet series on fractals.

Findings

Successfully estimates dimensions of intersections with limsup sets.

Applies method to compute multifractal spectra of lacunary wavelet series.

Overcomes limitations of classical Mass Transference Principles.

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 $\mathcal{I}$.