Iterates of dynamical systems on compact metrizable countable spaces

Abstract

Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup and $E(X,f{)}^{\ast }=E(X,f)\setminus \{f^n:n\in \mathbb{N}\}$. We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a phase space. We show that if $(X,f)$ is a dynamical system such that $X$ is a compact metric countable space and every accumulation point $X^{\prime }$ is periodic, then either each function of $E(X,f{)}^{\ast }$ is continuous or each function of $E(X,f{)}^{\ast }$ is discontinuous. We describe an example of a dynamical system $(X,f)$ where $X$ is a compact metric countable space, the orbit of each accumulation point is finite and $E(X,f{)}^{\ast }$ contains continuous and discontinuous functions.