Return Time Sets of Minimal Points for Countable Group Actions

Abstract

For a countably infinite discrete group $G$, we study subsets of $G$ that contain a neighbourhood return-time set of a minimal point in a $G$-flow $X$. For $e\in F\subseteq G$, we characterize this property by the existence of $e\in A\subseteq F$ such that $\operatorname{Int}_K(A):=\{t\in G:Kt\subseteq A\}$ is right syndetic for every finite $K\subseteq A$, and equivalently by the support of a minimal binary configuration in the Bernoulli shift $2^G$. Additional conditions on $\operatorname{Int}_K(A) \operatorname{Int}_K(A)^{-1}$ characterize realization by minimal flows that are weakly mixing of all finite orders or strongly mixing: for every such $K$, the former requires that for each finite $H\subseteq G$ there is $c\in G$ with $HcH\subseteq \operatorname{Int}_K(A) \operatorname{Int}_K(A) ^{-1}$, whereas the latter requires that $\operatorname{Int}_K(A)\operatorname{Int}_K(A)^{-1}$ be cofinite. We also construct, for every compact metrizable minimal $G$-flow, a free minimal, strongly mixing Cantor flow disjoint from it.