Weak Gibbs Measures for Local Homeomorphisms

TL;DR

This paper establishes the existence and uniqueness of weak Gibbs measures for a broad class of local homeomorphisms and potentials, leading to results on equilibrium states and large deviations in diverse dynamical systems.

Contribution

It extends Gibbs measure theory to non-uniform and complex dynamical systems, including attractors and partially hyperbolic horseshoes, using a novel non-uniform conformal-like approach.

Findings

Proves existence and uniqueness of weak Gibbs measures for local homeomorphisms.

Shows the uniqueness of equilibrium states derived from Gibbs measures.

Establishes a large deviations principle for the studied systems.

Abstract

We study a broad class of local homeomorphisms and continuous potentials, proving the existence and uniqueness of weak Gibbs measures. From the Gibbs property, we show the uniqueness of equilibrium states and derive a large deviations principle. Furthermore, we extend these results to a class of attractors that are semiconjugate to local homeomorphisms from our original setting. Our approach is based on non-uniform conformal-like property and applies to a wide range of topological dynamical systems, including non-uniformly expanding maps, zooming local homeomorphisms, attractors arising from solenoid-like constructions and certain families of partially hyperbolic horseshoes.