Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets

TL;DR

This paper investigates the multifractal structure of weak Gibbs measures, establishing estimates for topological pressure related to Birkhoff averages and large deviations, with implications for irregular sets in hyperbolic dynamics.

Contribution

It provides new estimates for topological pressure of irregular sets for systems with weak Gibbs measures, extending results to non-uniform hyperbolic systems and flows.

Findings

Topological pressure of irregular sets can be expressed via large deviations rate functions.

Most irregular sets have strictly smaller topological pressure than the entire system.

Extensions to hyperbolic flows and non-uniformly hyperbolic systems are achieved.

Abstract

In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $\phi$ is an H\"older continuous potential we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb R$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.