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 geo desic flow on a compact rank 1 manifold admits a uniquely determined i nvariant measure of maximal entropy. This generalizes previous work of R. Bowen and G. Margulis. As an application we show that the exponent ial growth rate of the set of singular closed geodesics is strictly sm aller than the topological entropy.