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

TL;DR

This paper investigates the dynamics of group actions on zero-dimensional spaces, establishing equivalences among minimality, orbit closure properties, and recurrence, and linking distality with equicontinuity.

Contribution

It introduces new equivalences for group actions on zero-dimensional spaces, connecting minimality, orbit closure, recurrence, and distality in a unified framework.

Findings

Minimal action on orbit closures is equivalent to closed orbit relation.

Recurrence concept aligns with equicontinuous group actions.

Distality is equivalent to equicontinuity in this setting.

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 $G = \langle S \rangle$. 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.