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

2022/03/16 by Dai, Xiongping
#37B05 #54H15 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2203.08466

Abstract

Let X be a zero-dimensional locally compact Hausdorff space not necessarily metric and G a compactly generated topological group not necessarily abelian or countable. We define recurrence at a point for any continuous action of G on X, 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\ni x↦ Gx∈ 2^X is continuous, are pairwise equivalent. Consequently, if this dynamics is distal, then it is equicontinuous.

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