Multi-invariant measures and subsets on nilmanifolds

2013/09/24 by Zhiren Wang, Wang, Zhiren
Mathematics · Physics and Astronomy · #37A15 #37A35 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1309.6223

openalex publication_date 2013/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Ratner's theorem on SL(2,R)-invariant measures
• Invariant measures for higher-rank hyperbolic abelian actions

Related

• On idempotent measure conjecture and decomposition of invariant measures
• Invariant measures for quasiperiodically forced circle homeomorphisms
• Corrections to ‘Invariant measures for higher-rank hyperbolic abelian actions’
• Symmetry of entropy in higher rank diagonalizable actions and measure\n classification
• Multiplicity structure of preimages of invariant measures under\n finite-to-one factor maps