Properties of equilibrium states for geodesic flows over manifolds without focal points

TL;DR

This paper establishes the uniqueness and ergodic properties of equilibrium states for geodesic flows on rank 1 manifolds without focal points, under certain potential conditions, advancing understanding of their statistical behavior.

Contribution

It proves the uniqueness of equilibrium states for specific potentials and provides criteria for the pressure gap condition on manifolds without focal points.

Findings

Unique equilibrium states exist under the pressure gap condition.

Equilibrium states exhibit ergodic properties like equidistribution and the K-property.

Criteria for continuous potentials to satisfy the pressure gap condition are provided.

Abstract

We prove that for closed rank 1 manifolds without focal points the equilibrium states are unique for H\"older potentials satisfying the pressure gap condition. In addition, we provide a criterion for a continuous potential to satisfy the pressure gap condition. Moreover, we derive several ergodic properties of the unique equilibrium states including the equidistribution and the K-property.