Regularity of entropy, geodesic currents and entropy at infinity

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.