Thermodynamic formalism for random non-uniformly expanding maps

TL;DR

This paper develops a thermodynamic formalism for a broad class of random, non-uniformly expanding maps, establishing existence, uniqueness, and statistical properties of equilibrium states without requiring Markov or uniform expansion structures.

Contribution

It introduces a novel thermodynamic framework for random non-uniformly expanding maps, including a variational principle and weak Gibbs measures, without relying on Markov structures.

Findings

Unique equilibrium states at high temperature with weak Gibbs property

Exponential decay of correlations for the equilibrium states

A variational principle for relative pressure in random dynamical systems

Abstract

We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered $C^1$-potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Carath\'eodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.