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

Let X be a locally compact zero-dimensional space, let S be an equicontinuous set of homeomorphisms such that 1 2 S D S-¹, and suppose that Gx is compact for each x 2 X, where G D hSi. 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 ^T_g2_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.