Bernoulli property for homogeneous systems

Abstract

Let $G$ be a semisimple Lie group with Haar measure $\mu$ and let $\Gamma$ be an irreducible lattice in $G$. For $g\in G$, we consider left translation $L_g$ acting on $(G\backslash\Gamma,\mu)$. We show that if $L_g$ is $K$ (which is equivalent to positive entropy of $L_g$) then $L_g$ is a Bernoulli automorphism. As a corollary, we also obtain analogous results for homogeneous flows.