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

### Colin D. Reid

University of Newcastle, Callaghan, Australia

## Abstract

Let $X$ be a locally compact zero-dimensional space, let $S$ be an equicontinuous set of homeomorphisms such that $1∈S=S_{−1}$, and suppose that $Gx$ is compact for each $x∈X$, 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 $F⊆G$ finite such that $⋂_{g∈F}g(U)$ 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.

## Cite this article

Colin D. Reid, Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces. Groups Geom. Dyn. 14 (2020), no. 2, pp. 413–425

DOI 10.4171/GGD/549