Multifractal formalism for inverse measures of random weak Gibbs measures

TL;DR

This paper investigates the multifractal properties of inverse measures derived from random weak Gibbs measures on fractal sets, extending deterministic multifractal formalism to stochastic dynamical systems.

Contribution

It develops a new framework for the multifractal analysis of inverse measures in random dynamical systems, generalizing known results from deterministic settings.

Findings

Established conditions for the validity of multifractal formalism in this context

Extended heterogeneous ubiquity results to random Gibbs measures

Provided a theoretical foundation for analyzing inverse measures in random dynamics

Abstract

Any Borel probability measure supported on a Cantor set of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures supported on the attractor associated with some $C^1$ random dynamics encoded by a random subshift of finite type, and expanding in the mean. The study requires, in particular, to develop in this context of random dynamics a suitable extension of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.