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)\{fn:n∈N}<inline-graphic></inline-graphic> 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 2ℵ0<inline-graphic></inline-graphic>. 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)|≤|X|<inline-graphic></inline-graphic>. For dynamical systems of the form (ω2+1,f)<inline-graphic></inline-graphic>, we show that if there is a point with a dense orbit, then all elements of E(ω2+1,f)<inline-graphic></inline-graphic> are continuous functions. We present several examples of dynamical systems which have a point with a dense orbit. Such systems provide examples where E(ω2+1,f)<inline-graphic></inline-graphic> and ω2+1<inline-graphic></inline-graphic> are homeomorphic but not algebraically homeomorphic, where ω2+1<inline-graphic></inline-graphic> is taken with the usual ordinal addition as semigroup operation. [ABSTRACT FROM AUTHOR]