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Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 123–183 | Cite as

Multi-invariant measures and subsets on nilmanifolds

  • Zhiren Wang
Article

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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Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Pennsylvania State UniversityUniversity ParkUSA

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