Characterizations of distality via weak 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${}^{\ast }$-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${}^{\ast }$-equicontinuous if and only if it is pairwise central${}^{\ast }$-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${}^{\ast }$-equicontinuous.

20 pages. To appear in Discrete and Continuous Dynamical Systems. arXiv admin note: text overlap with arXiv:2108.01271