Iterates of dynamical systems on compact metrizable countable spaces

Given a dynamical system (X,f)(X,f), we let E(X,f)E(X,f) denote its Ellis semigroup and E(X,f)⁎=E(X,f)∖{fn:n∈N}E(X,f)⁎=E(X,f)∖{fn:n∈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)(X,f) is a dynamical system such that X is a compact metric countable space and every accumulation point of X   is periodic, then either all functions of E(X,f)⁎E(X,f)⁎ are continuous or all functions of E(X,f)⁎E(X,f)⁎ are discontinuous. We describe an example of a dynamical system (X,f)(X,f) where X   is a compact metric countable space, the orbit of each accumulation point is finite and E(X,f)⁎E(X,f)⁎ contains both continuous and discontinuous functions.