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

TL;DR

This paper studies recurrence in zero-dimensional locally compact flows with compactly generated phase groups, establishing equivalences among various recurrence and minimality conditions, and exploring implications for product recurrence and equicontinuity.

Contribution

It introduces a new definition of recurrence for compactly generated para-topological groups acting on zero-dimensional spaces and proves key equivalences among recurrence, minimality, and continuity conditions.

Findings

Recurrence conditions are equivalent under compact orbit closures.

Pointwise product recurrence implies equicontinuity and regular almost periodicity.

Existence of non-trivial equicontinuous factors for certain distal flows.

Abstract

We define recurrence for a compactly generated para-topological group $G$ acting continuously on a locally compact Hausdorff space $X$ with $\dim X=0$, 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 pointwise product recurrent, then it is pointwise regularly almost periodic and equicontinuous; moreover, a distal, compact, and non-connected $G$-flow has a non-trivial equicontinuous pointwise regularly almost periodic factor.