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

TL;DR

This paper investigates the properties of the Ellis semigroup for homeomorphisms on compact metric countable spaces, establishing equivalences among equicontinuity, distality, and periodicity, and providing a new proof of Ellis's theorem.

Contribution

It proves the equivalence of equicontinuity, distality, and periodicity for such systems and offers a direct proof of Ellis's characterization of distal systems.

Findings

Equicontinuous systems are exactly those where every point is periodic.

Distal systems correspond to Ellis semigroups that are groups.

The paper provides a new proof of Ellis's theorem relating distality and Ellis semigroups.

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.