Uniqueness and Stability of Equilibrium States for Random Non-uniformly Expanding Maps

TL;DR

This paper develops a thermodynamical formalism for a class of random non-uniformly expanding maps, proving the existence, uniqueness, and continuous dependence of equilibrium states and topological pressure.

Contribution

It introduces a thermodynamical formalism for random non-uniformly expanding maps with small potential variation, establishing existence, uniqueness, and stability of equilibrium states.

Findings

Existence of equilibrium states for the class of maps.

Uniqueness of these equilibrium states.

Continuous variation of equilibrium states and pressure.

Abstract

We consider a robust class of random non-uniformly expanding local homeomorphisms and H\"older continuous potentials with small variation. For each element of this class we develop the Thermodynamical Formalism and prove the existence and uniqueness of equilibrium states among non-uniformly expanding measures. Moreover, we show that these equilibrium states and the random topological pressure vary continuously in this setting.