Pressure at infinity and strong positive recurrence in negative curvature

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 Poincaré 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 to be 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.