A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds

Abstract

We give a short proof of the unique ergodicity of the strong stable foliation of the geodesic flow on the frame bundle of a hyperbolic manifold admitting a finite measure of maximal entropy. Equivalently, let G = S0o(n, 1), $\Gamma$ \textless{} G be a discrete subgroup of G, and G = N AK the Iwasawa decomposition of G. If the geodesic flow on $\Gamma$\G admits a finite measure of maximal entropy, we prove that the action of N on $\Gamma$\G by right multiplication admits a unique invariant measure supported on points whose A-orbit does not diverge.