Bernoulli property for homogeneous systems

TL;DR

This paper proves that certain homogeneous systems with positive entropy are Bernoulli automorphisms, extending results to flows on homogeneous spaces and providing a deeper understanding of their ergodic properties.

Contribution

It establishes that left translations with positive entropy on homogeneous spaces are Bernoulli automorphisms, a significant advancement in ergodic theory of homogeneous systems.

Findings

Left translation with positive entropy is Bernoulli.

Results extend to homogeneous flows.

Provides new insights into ergodic properties of 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.