Equilibrium stability for non-uniformly hyperbolic systems

TL;DR

This paper establishes the stability and continuity of equilibrium states and topological pressure for a broad class of non-uniformly hyperbolic systems and potentials, including skew products.

Contribution

It proves equilibrium stability and pressure continuity for non-uniformly hyperbolic maps and potentials, and demonstrates the continuous variation of equilibrium states in these systems.

Findings

Equilibrium states depend continuously on dynamics and potentials.

Topological pressure is continuous with respect to system changes.

Existence of finitely many ergodic equilibrium states for certain systems.

Abstract

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic H\"older continuous potentials. Finally we show that these equilibrium states vary continuously in the weak$^\ast$ topology within such systems.