Measure of maximal entropy for H-flows on non-compact manifolds

TL;DR

This paper introduces H-flows on non-compact manifolds, establishes conditions for the existence of maximal entropy measures, and compares different entropy notions in non-compact dynamical systems.

Contribution

It defines H-flows, proves the existence of maximal entropy measures under strong positive recurrence, and analyzes entropy concepts in non-compact settings.

Findings

H-flows include geodesic flows on non-compact manifolds with negative curvature

Existence of maximal entropy measures under strong positive recurrence

Comparison of various entropy notions in non-compact dynamical systems

Abstract

In this work, we introduce a natural class of chaotic flows on non-compact manifolds, called H-flows, which includes geodesic flows on non-compact manifolds with pinched negative curvature. We show that, under the additional assumption, called strong positive recurrence, that their entropy at infinity is strictly smaller than the topological entropy, such flows admit an invariant probability measure maximizing entropy. In particular, we compare several notions of entropy in a non-compact setting.