Unique equilibrium states for flows and homeomorphisms with non-uniform structure

TL;DR

This paper extends the uniqueness of equilibrium states in dynamical systems to cases with weaker, non-uniform conditions, broadening the applicability of thermodynamic formalism.

Contribution

It demonstrates that unique equilibrium states persist under non-uniform versions of specification, expansivity, and the Bowen property for flows and homeomorphisms.

Findings

Unique equilibrium states under non-uniform conditions

Upper bounds from large deviations principles

Framework for thermodynamic formalism in complex systems

Abstract

Using an approach due to Bowen, Franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the Bowen property. We show that this conclusion remains true using weaker non-uniform versions of specification, expansivity, and the Bowen property. We also establish a corresponding result for homeomorphisms. In the homeomorphism case, we obtain the upper bound from the level-2 large deviations principle for the unique equilibrium state. The theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic formalism in partially hyperbolic and non-uniformly hyperbolic settings.