A zero-one law for dynamical properties

Abstract

For any countable group $\Gamma$ satisfying the ``weak Rohlin property'', and for any dynamical property, the set of $\Gamma$-actions with that property is either residual or meager. The class of groups with the weak Rohlin property includes each lattice $\integers^{\times{d}}$; indeed, all countable discrete amenable groups. For $\Gamma$ an arbitrary countable group, let $\actsp$ be the set of $\Gamma$-actions on the unit circle $Y$. We establish an Equivalence theorem by showing that a dynamical property is Baire/meager/residual in $\actsp$ if and only if it is Baire/meager/residual in the set of shift-invariant measures on the product space $Y^{\times\Gamma}$.