The thermodynamic approach to multifractal analysis

TL;DR

This paper extends the thermodynamic approach to multifractal analysis from uniformly hyperbolic systems to more general non-uniformly hyperbolic systems, including those with phase transitions, and compares it with the saturation approach.

Contribution

It broadens the applicability of the thermodynamic multifractal formalism to non-uniformly hyperbolic systems, unifying various results and including systems with phase transitions.

Findings

Extended the thermodynamic framework to non-uniformly hyperbolic systems.

Compared thermodynamic and saturation approaches in multifractal analysis.

Provided a survey of existing multifractal results in the literature.

Abstract

Most results in multifractal analysis are obtained using either a thermodynamic approach based on existence and uniqueness of equilibrium states or a saturation approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions. In the process, we compare this thermodynamic approach with the saturation approach and give a survey of many of the multifractal results in the literature.