Multi-invariant measures and subsets on nilmanifolds
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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
We also deduce a rigidity property for invariant closed subsets.
the factor measure has zero entropy under every element of the action
the factor action is virtually cyclic.
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