Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps

TL;DR

This paper establishes the existence, uniqueness, and stability of equilibrium states for a broad class of non-uniformly expanding maps on compact manifolds, without relying on Markov structures, and explores their statistical properties.

Contribution

It proves the existence and uniqueness of equilibrium states for non-uniformly expanding maps with Holder potentials, and demonstrates their stability under perturbations.

Findings

Existence of finitely many ergodic equilibrium states.

Uniqueness and exactness of the equilibrium state in topologically mixing cases.

Statistical stability and robustness under stochastic perturbations.

Abstract

We prove existence of finitely many ergodic equilibrium states for a large class of non-uniformly expanding local homeomorphisms on compact manifolds and Holder continuous potentials with not very large oscillation. No Markov structure is assumed. If the transformation is topologically mixing there is a unique equilibrium state, it is exact and satisfies a non-uniform Gibbs property. Under mild additional assumptions we also prove that the equilibrium states vary continuously with the dynamics and the potentials (statistical stability) and are also stable under stochastic perturbations of the transformation.