Characterizations of distality via weak equicontinuity

TL;DR

This paper characterizes distality in dynamical systems via various weak forms of equicontinuity, linking these properties to measure invariance, product systems, and the algebraic structure of the acting group.

Contribution

It introduces new weak equicontinuity notions and establishes their equivalence to distality under different conditions in minimal systems.

Findings

Distality is equivalent to pairwise IP*-equicontinuity in systems with invariant measures.

In systems with dense minimal points in the product, distality aligns with pairwise IP*- and central*-equicontinuity.

For abelian groups, systems of order infinity correspond to pairwise FIP*-equicontinuity.

Abstract

For an infinite discrete group $G$ acting on a compact metric space $X$, we introduce several weak versions of equicontinuity along subsets of $G$ and show that if a minimal system $(X,G)$ admits an invariant measure then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous; if the product system $(X\times X,G)$ of a minimal system $(X,G)$ has a dense set of minimal points, then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous if and only if it is pairwise central$^*$-equicontinuous; if $(X,G)$ is a minimal system with $G$ being abelian, then $(X,G)$ is a system of order $\infty$ if and only if it is pairwise FIP$^*$-equicontinuous.