# Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points

## TL;DR

This paper proves the uniqueness of the measure of maximal entropy for geodesic flows on certain closed surfaces without conjugate points, and discusses conditions for extending these results to higher dimensions.

## Contribution

It establishes the uniqueness and mixing properties of the measure of maximal entropy for geodesic flows on specific manifolds without conjugate points, and explores higher-dimensional extensions.

## Key findings

- Unique measure of maximal entropy exists for these geodesic flows.
- The measure is fully supported and the flow is mixing.
- Conditions for higher-dimensional generalizations are formulated.

## Abstract

We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.