Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces

Abstract

Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1\in S=S^{-1}$, and suppose that $\overline{Gx}$ is compact for each $x\in X$, where $G=⟨S⟩$. We show in this setting that a number of conditions are equivalent: (a) $G$ acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset $U$ of $X$, there is $F\subseteq G$ finite such that ${\bigcap }_{g\in F}g(U)$ is $G$-invariant. All of these are equivalent to a notion of recurrence, which is a variation on a concept of Auslander–Glasner–Weiss. It follows in particular that the action is distal if and only if it is equicontinuous.