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

We define recurrence for a compactly generated para-topological group G acting continuously on locally compact Hausdorff space X with dimX=0, and then we show that if Gx‾ is all x∈X, the conditions are pairwise equivalent: (i) this dynamic pointwise recurrent, (ii) union of G-minimal sets, (iii) G-orbit closure relation closed in X×X, (iv) X∋x↦Gx‾∈2X continuous. Consequently, product it regularly almost periodic equicontinuous; moreover, distal, compact, nonconnected G-flow has nontrivial equicontinuous factor.