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

Abstract

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

41 pages, 4 figures. Added a discussion of equidistribution results in Section 2.3; fixed the construction of the MME from the Patterson-Sullivan measure in Section 5.2; moved the proof of specification from an appendix to Section 4.2; fixed various typos and minor errors