Résumé
In this work, we introduce a notion of entropy at infinity for the geodesic flow of negatively curved manifolds. We introduce the class of noncompact manifolds 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 C-1-uniform perturbations of the metric. This result generalizes former work of Katok (1982) and Katok-Knieper-Weiss (1991) in the compact case.