The uniqueness of the measure of maximal entropy for geodesic flows on rank $1$ manifolds


In this paper we prove a conjecture of A. Katok, stating that the geodesic flow on a compact rank $1$ manifold admits a uniquely determined invariant measure of maximal entorpy. This generalizes previous work of R. Bowen and G. Margulis. As an application we show that the exponential growth rate of the set of singular closed geodesics is strictly smaller than the topological entropy.


Gerhard Knieper