# Equilibrium states for non-uniformly hyperbolic systems: statistical properties and analyticity

## TL;DR

This paper studies equilibrium states for non-uniformly hyperbolic systems, proving their existence, uniqueness, spectral properties, decay of correlations, and analyticity of thermodynamic quantities, with applications to skew products.

## Contribution

It establishes the spectral gap and analyticity of equilibrium states for a broad class of non-uniformly hyperbolic maps and potentials, extending thermodynamic formalism.

## Key findings

- Spectral gap for transfer operators in H"older spaces
- Exponential decay of correlations for equilibrium states
- Analytic dependence of thermodynamic quantities on potentials

## Abstract

We consider a wide family of non-uniformly expanding maps and hyperbolic H\"older continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in some space of H\"older continuous observables and from this we obtain an exponential decay of correlations and a central limit theorem for the equilibrium state. Moreover, we establish the analyticity with respect to the potential of the equilibrium state as well as that of other thermodynamic quantities. Furthermore, we derive similar results for the equilibrium state associated to a family of non-uniformly hyperbolic skew products and hyperbolic H\"older continuous potentials.