Regularity, linear response formula and differentiability of the free energy for non-uniformly expanding local homeomorphisms

TL;DR

This paper investigates equilibrium states for a class of non-uniformly expanding local homeomorphisms, establishing their existence, uniqueness, and the differentiability of related statistical quantities like free energy.

Contribution

It introduces conditions under which equilibrium states exist uniquely and proves the differentiability of free energy and equilibrium states with respect to the system.

Findings

Existence and uniqueness of equilibrium states.

Differentiability of free energy function.

Regularity results for statistical quantities.

Abstract

We study equilibrium states for an open class of non-uniformly expanding local homeomorphisms defined by a mild condition such that for some iterate each point admits at least one contracting inverse branch. We prove the existence and uniqueness of equilibrium states and the differentiability of statistical quantities (such as the equilibrium states and the free energy function) with respect to the dynamical system.