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

Let X be a compact metric countable space, let f : X → X be a homeomorphism and let E ( X , f ) be its Ellis semigroup. Among other results we show that the following statements are equivalent: (1) ( X , f ) is equicontinuous, (2) ( X , f ) is distal and (3) 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.