Beyond Bowen's Specification Property

TL;DR

This paper reviews recent advances in thermodynamic formalism that extend Bowen's classical results beyond uniformly hyperbolic systems, applying to partially hyperbolic systems and geodesic flows, with new criteria for equilibrium state uniqueness.

Contribution

It introduces generalized specification properties and a new criterion for equilibrium state uniqueness in partially hyperbolic systems with 1-dimensional center.

Findings

Extended Bowen's arguments to non-uniform hyperbolic systems

Established new criteria for equilibrium state uniqueness

Applied results to geodesic flows beyond negative curvature

Abstract

A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every H\"older continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowen's arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for partially hyperbolic systems with 1-dimensional center.