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

Abstract

We define recurrence for a compactly generated para-topological group G acting continuously on a locally compact Hausdorff space X with $dimX=0$, and then we show that if $\overline{Gx}$ is compact for all $x\in X$, the conditions are pairwise equivalent: (i) this dynamic 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. Consequently, if this dynamic is pointwise product recurrent, then it is pointwise regularly almost periodic and equicontinuous; moreover, a distal, compact, and nonconnected G-flow has a nontrivial equicontinuous pointwise regularly almost periodic factor.