Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

Gerhard Knieper

TL;DR

The paper addresses the uniqueness of the measure of maximal entropy (MME) for geodesic flows on closed manifolds without conjugate points under coarse hyperbolicity, assuming a Gromov hyperbolic, residually finite fundamental group and a divergence property. It constructs an MME via Patterson–Sullivan measures on the boundary, yielding a ν-ergodic, mixing measure μ with full support and μ(ℰ)=1, under entropy-gap or interior-expansiveness conditions. The main contribution is proving the MME is unique under these conditions, using entropy-expansiveness techniques, partition refinements, and lift-to-cover arguments to connect to finite covers with large injectivity radius. This extends previous results that relied on negative curvature, providing a robust framework for uniqueness in the absence of conjugate points and broad coarse hyperbolicity assumptions, with implications for counting closed geodesics and the statistical behavior of geodesic flows.

Abstract

In this article we study geodesic flows on closed Riemannian manifolds without conjugate points and divergence property of geodesic rays. If the fundamental group is Gromov hyperbolic and residually finite we prove, under appropriate assumptions on the expansive set, that the geodesic flow has a unique measure of maximal entropy. This generalizes corresponding results of Climenhaga, Knieper and War proved under the stronger assumption of the existence of a background metric of negative sectional curvature.

Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points

TL;DR

The paper addresses the uniqueness of the measure of maximal entropy (MME) for geodesic flows on closed manifolds without conjugate points under coarse hyperbolicity, assuming a Gromov hyperbolic, residually finite fundamental group and a divergence property. It constructs an MME via Patterson–Sullivan measures on the boundary, yielding a ν-ergodic, mixing measure μ with full support and μ(ℰ)=1, under entropy-gap or interior-expansiveness conditions. The main contribution is proving the MME is unique under these conditions, using entropy-expansiveness techniques, partition refinements, and lift-to-cover arguments to connect to finite covers with large injectivity radius. This extends previous results that relied on negative curvature, providing a robust framework for uniqueness in the absence of conjugate points and broad coarse hyperbolicity assumptions, with implications for counting closed geodesics and the statistical behavior of geodesic flows.

Abstract

In this article we study geodesic flows on closed Riemannian manifolds without conjugate points and divergence property of geodesic rays. If the fundamental group is Gromov hyperbolic and residually finite we prove, under appropriate assumptions on the expansive set, that the geodesic flow has a unique measure of maximal entropy. This generalizes corresponding results of Climenhaga, Knieper and War proved under the stronger assumption of the existence of a background metric of negative sectional curvature.

Paper Structure

This paper contains 13 sections, 28 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Gromov hyperbolic manifolds without conjugate points
  3. Basic facts on Gromov hyperbolic metric spaces
  4. Gromov hyperbolic groups
  5. Closed manifolds without conjugate points and Gromov hyperbolic fundamental groups
  6. Patterson--Sullivan measure and the MME
  7. Poincaré series and the Patterson--Sullivan measure
  8. Construction of the measure of maximal entropy
  9. Entropy expansiveness
  10. The uniqueness of the measure of maximal entropy
  11. Properties of the measure $\mu$ and partitions
  12. Proof of Theorem \ref{['thm:uniqueMME-inj']}
  13. Proof of uniqueness of the MME

Key Result

Theorem 1

Let $(M,g)$ be a closed smooth $(C^\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group. Assume that for all non-expansive measures $\nu \in \mathcal{M}_\phi(SM)$. Then the geodesic flow has a unique measure $\mu$ of maximal entropy. Furthermore, $\mu(\mathcal{E}) =1$, $\mu$ is mixing and full

Theorems & Definitions (69)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • ...and 59 more