The Graph and Range Singularity Spectra of Random Wavelet Series built from Gibbs measures

TL;DR

This paper investigates the multifractal properties of random wavelet series derived from Gibbs measures, focusing on the singularity spectra of their graphs and ranges on iso-Hölder sets, using Gibbs measures on subshifts of finite type.

Contribution

It introduces a novel method to analyze the singularity spectra of wavelet series built from Gibbs measures, linking multifractal analysis with symbolic dynamics.

Findings

Determined the singularity spectra for the graph and range of these wavelet series.

Established a connection between Gibbs measures on subshifts and the multifractal structure of wavelet functions.

Provided a framework for analyzing multifractality in wavelet series using symbolic dynamics.

Abstract

We consider multifractal random wavelet series built from Gibbs measures, and study the singularity spectra associated with the graph and range of these functions restricted to their iso-H\"older sets. To obtain these singularity spectra, we use a family of Gibbs measures defined on a sequence of topologically transitive subshift of finite type whose Hausdorff distance to the set of zeros of the mother wavelet tends to 0.