The McMahon pseudo-metrics of minimal semiflows with invariant measures

TL;DR

This paper explores the properties of minimal semiflows with invariant measures using McMahon pseudo-metrics, establishing relationships between key dynamical relations and characterizing weak-mixing, especially in almost automorphic cases.

Contribution

It introduces new characterizations of minimal semiflows with invariant measures using McMahon pseudo-metrics and extends Veech's Structure Theorem to almost automorphic semiflows.

Findings

Established relationships between equicontinuous, proximal, and Veech relations.

Characterized weak-mixing in terms of McMahon pseudo-metrics.

Extended Veech's Structure Theorem to almost automorphic semiflows.

Abstract

Using McMahon pseudo-metrics, for any minimal semiflow admitting an invariant measure, we study the relationships between its equicontinuous structure relation, regionally proximal relation and Veech's relations; and characterize its weak-mixing. We show its Veech Structure Theorem if it is almost automorphic.