On the Ellis semigroup of a cascade on a compact metric countable space

Abstract

Let $X$ be a compact metric countable space, let $f:X\to X$ be a homeomorphism and let $E(X,f)$ be its Ellis semigroup. Among other results we show that the following statements are equivalent: (i) $(X,f)$ is equicontinuous, (ii) $(X,f)$ is distal and (iii) every point is periodic. We use this result to give a direct proof of a theorem of Ellis saying that $(X,f)$ is distal if, and only if, $E(X,f)$ is a group.