Equilibrium states in negative curvature

TL;DR

This paper develops a new framework using Patterson-Sullivan densities to study Gibbs measures on negatively curved manifolds without compactness assumptions, leading to various existence, uniqueness, and finiteness results.

Contribution

It introduces a framework that removes compactness constraints, enabling broader analysis of Gibbs measures and their applications in ergodic theory and geometric dynamics.

Findings

Established existence and uniqueness of Gibbs measures under new conditions

Proved finiteness results for Gibbs measures on non-compact manifolds

Applied results to orbit counting, equidistribution, and ergodic properties

Abstract

With their origin in thermodynamics and symbolic dynamics, Gibbs measures are crucial tools to study the ergodic theory of the geodesic flow on negatively curved manifolds. We develop a framework (through Patterson-Sullivan densities) allowing us to get rid of compactness assumptions on the manifold, and prove many existence, uniqueness and finiteness results of Gibbs measures. We give many applications, to the Variational Principle, the counting and equidistribution of orbit points and periods, the unique ergodicity of the strong unstable foliation and the classification of Gibbs densities on some Riemannian covers.