Cardinality of the Ellis semigroup on compact metric countable spaces

Abstract

Let $E(X,f)$ be the Ellis semigroup of a dynamical system $(X,f)$ where $X$ is a compact metric space. We analyze the cardinality of $E(X,f)$ for a compact countable metric space $X$. A characterization when $E(X,f)$ and $E(X,f)^* = E(X,f) \setminus \{ f^n : n \in \mathbb{N}\}$ are both finite is given. We show that if the collection of all periods of the periodic points of $(X,f)$ is infinite, then $E(X,f)$ has size $2^{\aleph_0}$. It is also proved that if $(X,f)$ has a point with a dense orbit and all elements of $E(X,f)$ are continuous, then $|E(X,f)| \leq |X|$. For dynamical systems of the form $(\omega^2 +1,f)$, we show that if there is a point with a dense orbit, then all elements of $E(\omega^2+1,f)$ are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where $E(\omega^2+1,f)$ and $\omega^2+1$ are homeomorphic but not algebraically homeomorphic, where $\omega^2+1$ is taken with the usual ordinal addition as semigroup operation.