On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points

Abstract

<p style='text-indent:20px;'>In this article, we consider the geodesic flow on a compact rank <inline-formula><tex-math id="M2">\begin{document}$ 1 $\end{document}</tex-math></inline-formula> Riemannian manifold <inline-formula><tex-math id="M3">\begin{document}$ M $\end{document}</tex-math></inline-formula> without focal points, whose universal cover is denoted by <inline-formula><tex-math id="M4">\begin{document}$ X $\end{document}</tex-math></inline-formula>. On the ideal boundary <inline-formula><tex-math id="M5">\begin{document}$ X(\infty) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M6">\begin{document}$ X $\end{document}</tex-math></inline-formula>, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based on the the Patterson-Sullivan measure, we show that the geodesic flow on <inline-formula><tex-math id="M7">\begin{document}$ M $\end{document}</tex-math></inline-formula> has a unique invariant measure of maximal entropy. We also obtain the asymptotic growth rate of the volume of geodesic spheres in <inline-formula><tex-math id="M8">\begin{document}$ X $\end{document}</tex-math></inline-formula> and the growth rate of the number of closed geodesics on <inline-formula><tex-math id="M9">\begin{document}$ M $\end{document}</tex-math></inline-formula>. These results generalize the work of Margulis and Knieper in the case of negative and nonpositive curvature respectively.