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 Hausdor ff space X with dim X = 0, and then, show that if Gx is compact for all x ∈ 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 × X , and (iv) X ∋ x 7→ Gx ∈ 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.