Equilibrium states for non-uniformly expanding maps: decay of correlations and strong stability

TL;DR

This paper investigates the decay of correlations and stability of equilibrium states in non-uniformly expanding maps, establishing spectral gaps and exponential decay without Markov assumptions, and analyzing stability under perturbations.

Contribution

It provides new results on spectral gaps, decay rates, and stability of equilibrium states for a broad class of non-uniformly expanding maps without Markov assumptions.

Findings

Spectral gap for Ruelle-Perron-Frobenius operator on Hölder spaces.

Exponential decay of correlations and proof of the central limit theorem.

Strong stability of equilibrium states under deterministic and random perturbations.

Abstract

We study the rate of decay of correlations for equilibrium states associated to a robust class of non-uniformly expanding maps where no Markov assumption is required. We show that the Ruelle-Perron-Frobenius operator acting on the space of Holder continuous observables has a spectral gap and deduce the exponential decay of correlations and the central limit theorem. In particular, we obtain an alternative proof for the existence and uniqueness of the equilibrium states and we prove that the topological pressure varies continuously. Finally, we use the spectral properties of the transfer operators in space of differentiable observables to obtain strong stability results under deterministic and random perturbations.