Pressure at infinity and strong positive recurrence in negative curvature

TL;DR

This paper explores entropy and pressure at infinity in negatively curved manifolds, establishing their equivalence and linking strong positive recurrence of potentials to the existence of finite Gibbs measures.

Contribution

It introduces three equivalent definitions of entropy and pressure at infinity and connects strong positive recurrence with finite Gibbs measures in this geometric setting.

Findings

Entropy and pressure at infinity coincide.

Strong positive recurrence implies existence of finite Gibbs measure.

Provides criteria and examples for strongly positively recurrent potentials.

Abstract

In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincar\'e series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show in particular that if a potential is strongly positively recurrent, then it admits a finite Gibbs measure. We also provide easy criteria allowing to build such strong positively recurrent potentials and many examples.