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 Gx‾ is compact for all x∈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×X, and (iv) X∋x↦Gx‾∈2X 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.