On the patterson-sullivan measure for geodesic flows on rank 1 manifolds without focal points

Abstract

In this article, we consider the geodesic ow on a compact rank 1 Riemannian manifold M without focal points, whose universal cover is denoted by X. On the ideal boundary X(1) of X, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic ow on M has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in X and the growth rate of the number of closed geodesics on M. These results generalize the work of Margulis and Knieper in the case of negative and nonpositive curvature respectively.