Uniqueness of the measure of maximal entropy for geodesic flows on surfaces

TL;DR

This paper proves the uniqueness of the measure of maximal entropy for transitive geodesic flows with positive topological entropy on closed surfaces, extending previous results and including new examples, and also shows at most one SRB measure exists.

Contribution

It establishes the uniqueness of the measure of maximal entropy for a broad class of geodesic flows on surfaces, covering previous cases and new examples.

Findings

Uniqueness of the measure of maximal entropy is proven for transitive geodesic flows with positive entropy.

The result applies to new examples constructed by Donnay and Burns-Donnay.

At most one SRB measure exists in this setting.

Abstract

We prove that if a geodesic flow on a closed orientable $C^\infty$ surface is transitive and has positive topological entropy, then it has a unique measure of maximal entropy. This covers all previous results of the literature on the uniqueness of the measure of maximal entropy in this context, as well as it applies to new examples such as the ones constructed by Donnay and Burns-Donnay. We also prove that, in the above context, there is at most one SRB measure.