Ruelle's inequality in negative curvature

TL;DR

This paper extends classical entropy concepts to noncompact Riemannian manifolds, establishing Ruelle's inequality and Pesin's formula for geodesic flows in negatively curved spaces.

Contribution

It introduces a new entropy notion for noncompact manifolds and proves fundamental inequalities for geodesic flows with negative curvature.

Findings

Established a new entropy concept for noncompact spaces

Proved Ruelle's inequality for geodesic flows in negatively curved manifolds

Validated Pesin's entropy formula in this setting

Abstract

In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.