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

2025/11/27 by Yuri Lima, Lima, Yuri, Davi Obata +3
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2511.22022

Abstract

We prove that if a geodesic flow on a closed orientable C^∞ 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.

Citations

• Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points
• Symbolic dynamics for non-uniformly hyperbolic flows in high dimension
• Homoclinic classes for flows: ergodicity and SRB measures
• From curve shortening to flat link stability and Birkhoff sections of geodesic flows
• On ergodic properties of geodesic flows on uniform visibility manifolds without conjugate points
• Homoclinic classes of geodesic flows on rank 1 manifolds
• Expansive factors for geodesic flows of compact manifolds without conjugate points and with visibility universal covering
• Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows
• Generic properties of 3-dimensional Reeb flows: Birkhoff sections and entropy
• Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points
• Measures of maximal entropy for surface diffeomorphisms
• Unique equilibrium states for geodesic flows in nonpositive curvature
• A Criterion For Ergodicity of Non-uniformly hyperbolic Diffeomorphisms

Related

• Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points
• Uniqueness of the measure of maximal entropy for geodesic flows on coarse hyperbolic manifolds without conjugate points
• Measures of maximal entropy for C^∞ three-dimensional flows
• The unique measure of maximal entropy for a compact rank one locally\n CAT(0) space
• Measure of maximal entropy for H-flows on non-compact manifolds