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Multi-invariant measures and subsets on nilmanifolds

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Abstract

Given a Zr-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. (1)

    the factor measure has zero entropy under every element of the action

  2. (2)

    the factor action is virtually cyclic.

We also deduce a rigidity property for invariant closed subsets.

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Authors and Affiliations

  1. Pennsylvania State University, University Park, PA, 16802, USA

    Zhiren Wang

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  1. Zhiren Wang
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Correspondence to Zhiren Wang.

Additional information

The author was supported by NSF grant DMS-1201453 and an AMS-Simons travel grant.

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Wang, Z. Multi-invariant measures and subsets on nilmanifolds. JAMA 135, 123–183 (2018). https://doi.org/10.1007/s11854-018-0041-z

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  • DOI: https://doi.org/10.1007/s11854-018-0041-z

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