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 = \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.