Volume asymptotics and Margulis function in nonpositive curvature

In this article, we consider a closed rank one $C^infty$ 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 $xin X$, and $h$ the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates [lim_{tto infty}b_t(x)/frac{e^{ht}}{h}=c(x)] for some continuous function $c: Xto mathbb{R}$. We prove that the Margulis function $c(x)$ is in fact $C^1$. If $M$ is a surface of nonpositive curvature without flat strips, we show that $c(x)$ is constant if and only if $M$ has constant negative curvature.