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arXiv · 2501.17712

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

Céline Esser·Béatrice Vedel
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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 $\mathcal{I}$.

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Céline Esser, Béatrice Vedel. 2025-01-29. Constructing self-similar subsets within the fractal support of Lacunary Wavelet Series for their multifractal analysis. https://arxiv.org/abs/2501.17712

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