Let E(X, f) be the Ellis semigroup of a discrete discrete dynamical system (X, f) where X is a compact countable metric space. Set \(P(X,f): = \{ |\mathcal {O}_f(x)|: x \ \text {is a periodic point of} \ X \}\) , for an eventually periodic point \(x\in X\) let \(l_x \in \mathbb N\) be its waiting time, and set \(L(X,f):= \{ l \in \mathbb {N}: \exists x \in X(x \ \text {is eventually periodic and} \ l = l_x) \}\) . We give two necessary and sufficient statements equivalent to the assertion “E(X, f) is a compactification of \(\mathbb N\) with the discrete topology”. And we also prove the following assertions: If each \(x\in X\) has a finite orbit and L(X, f) and P(X, f) are finite, then \( |E(X,f)| \le \prod _{s\in P(X,f)}\{0,\dots , s-1\} + \max L(X,f). \) Besides, if all the periods are relative prime numbers, then \( |E(X,f)|= \prod _{s\in P(X,f)}\{0,\dots , s-1\}+\max L(X,f). \) Two necessary conditions on a discrete dynamical system are given in order that its Ellis semigroup be countable. We also include several examples that exemplify the diversity of dynamical properties in relation to the cardinality of the Ellis semigroup.
