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

Title: Equicontinuity, orbit closures and invariant compact open sets for group actions on zero-dimensional spaces
Creator: Reid, Colin D.
Relation: ARC.DP120100996 http://purl.org/au-research/grants/arc/DP120100996
Relation: Groups, Geometry, and Dynamics Vol. 14, Issue 2, p. 413-425
Publisher Link: http://dx.doi.org/10.4171/GGD/549
Publisher: E M S Press
Resource Type: journal article
Date: 2020
Description: Let XXX be a locally compact zero-dimensional space, let SSS be an equicontinuous set of homeomorphisms such that 1∈S=S−11 \in S = S^{-1}1∈S=S−1, and suppose that Gx‾\overline{Gx}Gx is compact for each x∈Xx \in Xx∈X, where G=⟨S⟩G = \langle S \rangleG=⟨S⟩. We show in this setting that a number of conditions are equivalent: (a) GGG acts minimally on the closure of each orbit; (b) the orbit closure relation is closed; (c) for every compact open subset UUU of XXX, there is F⊆GF \subseteq GF⊆G finite such that ⋂g∈Fg(U)\bigcap_{g \in F}g(U)⋂g∈Fg(U) is GGG-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.
Subject: cantor dynamics; finitely generated group actions; recurrence; topological group theory
Identifier: http://hdl.handle.net/1959.13/1440634
Identifier: uon:41202
Identifier: ISSN:1661-7207
Language: eng
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