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

2017/10/02 by Reid, Colin D.
#Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1710.00627

Abstract

Let X be a locally compact zero-dimensional space, let S be an equicontinuous set of homeomorphisms such that 1 ∈ S = S^-1, and suppose that Gx is compact for each x ∈ X, where G = ⟨ S ⟩. 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 \bigcap_{g ∈ 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.

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