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

TL;DR

The paper establishes that transitive geodesic flows on closed orientable surfaces with positive topological entropy have a unique measure of maximal entropy, which is Bernoulli; under an additional full-support positive-entropy invariant measure, this MME has full support. The authors develop a unified framework combining symbolic coding of homoclinic classes with Birkhoff sections to reduce the problem to 2D return maps and apply equilibrium-state uniqueness results, extending to general flows on 3-manifolds and Hölder potentials. This yields a broad uniqueness principle that encompasses earlier results and newly constructed examples, and also proves at most one SRB measure in this setting. Moreover, when the Liouville measure is hyperbolic, it is Bernoulli, linking geometric entropy to strong statistical properties of the flow.

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.

Figures (2)

• Figure 1: A metric on the sphere $\mathbb S^2$ whose geodesic flow is ergodic and hyperbolic for the Liouville measure Burns-Donnay.
• Figure 2: Construction of $su$--quadrilateral.

Theorems & Definitions (24)

• Theorem A
• Corollary 1.1
• proof
• Theorem B
• Theorem C
• Theorem D
• Lemma 2.2
• proof
• Lemma 2.3
• proof