Title: 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.