On recurrence in zero-dimensional locally compact flow with compactly generated phase group

TL;DR

This paper studies recurrence in zero-dimensional locally compact spaces under actions of compactly generated topological groups, establishing equivalences among various recurrence and minimality conditions, and showing distal actions are equicontinuous.

Contribution

It introduces a new framework for recurrence in non-metric zero-dimensional spaces under general group actions and proves key equivalences among recurrence, minimality, and continuity properties.

Findings

Recurrence at points is characterized for actions of compactly generated groups.

Equivalence of recurrence, minimality, closed orbit relations, and continuity of orbit closures.

Distal actions are shown to be equicontinuous in this setting.

Abstract

Let $X$ be a zero-dimensional locally compact Hausdorff space not necessarily metric and $G$ a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of $G$ on $X$, and then, show that if $\overline{Gx}$ is compact for all $x\in X$, the conditions (i) this dynamics is pointwise recurrent, (ii) $X$ is a union of $G$-minimal sets, (iii) the $G$-orbit closure relation is closed in $X\times X$, and (iv) $X\ni x\mapsto \overline{Gx}\in 2^X$ is continuous, are pairwise equivalent. Consequently, if this dynamics is distal, then it is equicontinuous.