Disjointness with all minimal systems under group actions

TL;DR

This paper characterizes when a transitive group action system is disjoint from all minimal systems, linking disjointness to infinite transitivity and the density of minimal points, with implications for systems with dense distal points.

Contribution

It provides a necessary and sufficient condition for disjointness with all minimal systems under group actions, connecting disjointness to infinite transitivity and minimal point density.

Findings

Disjointness implies infinite transitivity and dense minimal points.

Any infinite transitive system with dense distal points is disjoint from all minimal systems.

Characterization of disjointness conditions for transitive systems under group actions.

Abstract

Let $G$ be a countable discrete group. We give a necessary and sufficient condition for a transitive $G$-system to be disjoint with all minimal $G$-systems, which implies that if a transitive $G$-system is disjoint with all minimal $G$-systems, then it is $\infty$-transitive, i.e. $(X^k,G)$ is transitive for all $k\in\N$, and has dense minimal points. In addition, we show that any $\infty$-transitive $G$-system with dense distal points are disjoint with all minimal $G$-systems.