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

TL;DR

This paper provides a concise proof of the unique ergodicity of the strong stable foliation of the geodesic flow on hyperbolic manifolds with finite maximal entropy measure, highlighting a key property of the flow's invariant measures.

Contribution

It offers a simplified proof of unique ergodicity for the stable foliation on hyperbolic manifolds with finite entropy measure, extending understanding of invariant measures in this setting.

Findings

Proves the unique ergodicity of the strong stable foliation.

Shows the action of N admits a unique invariant measure.

Identifies the measure supported on non-diverging A-orbits.

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.