Regularity of entropy, geodesic currents and entropy at infinity

Barbara Schapira (1) , Samuel Tapie (2, 3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LMJL - Laboratoire de Mathématiques Jean Leray
3 IECL - Institut Élie Cartan de Lorraine

In this work, we introduce the notion of entropy at infinity, and define a wide class of noncompact manifolds with negative curvature, those which admit a critical gap between entropy at infinity and topological entropy. We call them strongly positively recurrent manifolds (SPR), and provide many examples. We show that dynamically, they behave as compact manifolds. In particular, they admit a finite measure of maximal entropy. Using the point of view of currents at infinity, we show that on these SPR manifolds the topological entropy of the geodesic flow varies in a C 1 -way along (uniformly) C 1 -perturbations of the metric. This result generalizes former work of Katok (1982) and Katok-Knieper-Weiss (1991) in the compact case.

Mots clés: strongly positively recurrent manifold; geodesic current; measure of maximal entropy; entropy at infinity; variation of entropy

Domaines: Systèmes dynamiques [math.DS]

Barbara Schapira, Samuel Tapie. Regularity of entropy, geodesic currents and entropy at infinity. Annales Scientifiques de l'École Normale Supérieure, 2021, quatrième série, 54 (1), pp.1--68. ⟨10.48550/arXiv.1802.04991⟩. ⟨hal-01708495v2⟩