Multi-invariant measures and subsets on nilmanifolds

Zhiren Wang

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arXiv · 1309.6223

Multi-invariant measures and subsets on nilmanifolds

Zhiren Wang

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Abstract

Given a $\mathbb Z^r$-action $α$ on a nilmanifold $X$ by automorphisms and an ergodic $α$-invariant probability measure $μ$, we show that $μ$ 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.

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BibTeX RIS

Zhiren Wang. 2013-09-24. Multi-invariant measures and subsets on nilmanifolds. https://arxiv.org/abs/1309.6223

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