Pressure at infinity and strong positive recurrence in negative curvature

Sébastien Gouëzel (1) , Camille Noûs (2) , Felipe Riquelme (3) , Barbara Schapira (1) , Samuel Tapie (4, 5)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Laboratoire Cogitamus = Cogitamus Laboratory
3 PUCV - Pontificia Universidad Católica de Valparaíso
4 IECL - Institut Élie Cartan de Lorraine
5 LMJL - Laboratoire de Mathématiques Jean Leray

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

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

Sébastien Gouëzel, Camille Noûs, Felipe Riquelme, Barbara Schapira, Samuel Tapie. Pressure at infinity and strong positive recurrence in negative curvature. Commentarii Mathematici Helvetici, In press. ⟨hal-02901142v2⟩