Regularity of entropy, geodesic currents and entropy at infinity

TL;DR

This paper introduces the concept of entropy at infinity for noncompact negatively curved manifolds, defines strongly positively recurrent manifolds (SPR), and shows that their topological entropy varies smoothly under metric perturbations, similar to compact cases.

Contribution

It defines entropy at infinity and introduces SPR manifolds, demonstrating their compact-like dynamical properties and smooth entropy variation under metric changes.

Findings

SPR manifolds admit a finite measure of maximal entropy

Topological entropy varies in a C¹ manner under metric perturbations on SPR manifolds

SPR manifolds exhibit behavior similar to compact manifolds in dynamics

Abstract

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.