Semicontinuity of measure theoretic entropy for noncompact systems

TL;DR

This paper investigates the semicontinuity of measure theoretic entropy for geodesic flows on noncompact Riemannian manifolds, exploring the escape of mass phenomenon and conditions for maximal entropy measures.

Contribution

It establishes upper semicontinuity of entropy in certain noncompact settings and provides criteria linking escape of mass to entropy, including for countable Markov shifts.

Findings

Proves upper semicontinuity of measure theoretic entropy for geodesic flows.

Provides criteria relating escape of mass to measure theoretic entropy.

Results applicable to nonpositively curved manifolds and countable Markov shifts.

Abstract

We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass phenomenon and the measure theoretic entropy on finite volume nonpositively curved manifolds satisfying the Visibility axiom. We provide a general criterion for the same relation to hold between the escape of mass and the measure theoretic entropy. This gives a criterion for the existence of measures of maximal entropy for the geodesic flow on some nonpositively curved manifolds. Finally, we prove some results in the context of countable Markov shifts.