Pressure at infinity and strong positive recurrence in negative curvature

Sébastien Gouëzel (1) , Camille Noûs (2) , Barbara Schapira (3) , Samuel Tapie (1) , Felipe Riquelme (4)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 Laboratoire Cogitamus
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 PUCV - Pontificia Universidad Católica de Valparaíso

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.

Mots clés: Keywords : Entropy at infinity; pressure at infinity; strong positive recurrence; noncompact negatively curved manifolds; thermodynamic formalism

Domaines: Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]

Sébastien Gouëzel, Camille Noûs, Barbara Schapira, Samuel Tapie, Felipe Riquelme. Pressure at infinity and strong positive recurrence in negative curvature. 2020. ⟨hal-02901142v1⟩