Characterizations of distality via weak equicontinuity on compact Hausdorff spaces

For an infinite discrete group $G$ acting on a compact Hausdorff space $X$, we characterize distality via weak equicontinuity introduced by Li and Yang. In other words, we 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 and $G$ is countable, then $(X,G)$ is distal if and only if it is pairwise IP$^*$-equicontinuous if and only if it is pairwise central$^*$-equicontinuous. This is a generalization of compact metric space of Li and Yang (Discrete Contin. Dyn. Syst. {\bf 44} (2024), no. 1, 61-77, DOI: 10.3934/dcds.2023096). Moreover, we provide a counterexample to illustrate that the dichotomy theorem for minimal systems regarding almost pairwise IP$^*$-equicontinuity does not hold in the non-metrizable case.