JournalsggdVol. 14, No. 2pp. 413–425

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

  • Colin D. Reid

    University of Newcastle, Callaghan, Australia
Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces cover
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Abstract

Let XX be a locally compact zero-dimensional space, let SS be an equicontinuous set of homeomorphisms such that 1S=S11 \in S = S^{-1}, and suppose that Gx\overline{Gx} is compact for each xXx \in X, where G=SG = \langle S \rangle. We show in this setting that a number of conditions are equivalent: (a) GG acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset UU of XX, there is FGF \subseteq G finite such that gFg(U)\bigcap_{g \in F}g(U) is GG-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