Cardinality of the Ellis semigroup on compact metric countable spaces

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 \(|E(X,f)| \le |X|\). 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.