Volume asymptotics and Margulis function in nonpositive curvature

Abstract

In this article, we consider a closed rank one Riemannian manifold M of nonpositive curvature and its universal cover X. Let b t (x) be the Riemannian volume of the ball of radius t > 0 around x ∈ X, and h the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates lim t→∞ b t (x)/e ht = c(x) for some continuous function c : X → R.