Multi-invariant measures and subsets on nilmanifolds

Abstract

Given a ${\mathbb{Z}}^r$-action $\alpha$ on a nilmanifold $X$ by automorphisms and an ergodic $\alpha$-invariant probability measure $\mu$, we show that $\mu$ is the uniform measure on $X$, unless modulo finite index modification, one of the following obstructions occurs for an algebraic factor action: 1. The factor measure has zero entropy under every element of the action; 2. The factor action is virtually cyclic. We also deduce a rigidity property for invariant closed subsets.

63 pages

References in corpus (1)

• Ratner's theorem on SL(2,R)-invariant measures