Type: Article

On the Ellis semigroup of a discrete dynamical system ([0, 1], f)

García-Ferreira, S., Vidal-Escobar, I.;

Abstract:

The Ellis semigroup of a discrete dynamical system (X, f) is denoted by E(X, f). We only consider surjective continuous functions f:[0,1]?[0,1] such that {fn:n?N} is infinite. The main result asserts that the Ellis semigroup E([0, 1], f) is always a compactification of the natural numbers with the discrete topology. We also prove some results estimating the cardinality of the Ellis semigroup of certain dynamical systems: we show that if f:[0,1]?[0,1] is a continuous function with positive topological entropy, then |E([0,1],f)|=22?0, and if f:[0,1]?[0,1] is a 2n-function for some n?N, then |E([0,1],f)?{fn:n?N}|=2n.