On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points

Abstract

If $(M,g)$ is a smooth compact rank $1$ Riemannian manifold without focal points, it is shown that the measure ${\mu }_{\max }$ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure ${\mu }_{\max }$ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $SM$ with respect to ${\mu }_{\max }$ is Bernoulli is acquired provided $M$ is a compact surface with genus greater than one and no focal points.

19 pages, 1 figures