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

2022/07/25 by Xiongping Dai, Dai, Xiongping
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN) #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2207.11852

openalex publication_date 2022/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define recurrence for a compactly generated para-topological group G acting continuously on a locally compact Hausdorff 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\ni x↦ 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.

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