A note on dynamics in functional spaces

Abstract

In this note, we study topologically transitive and hypercyclic composition operators on Cp(X) or C k (X) .We prove that if G is a semigroup of continuous self maps of a countable metric space X with the following properties:(1) every element of G is one-to-one on X , (2) the action of G is strongly run-away on X , then the action ofĜ on Cp(X) is topologically transitive and hypercyclic. If G is the set of all one-to-one and continuous self maps of R \ Z , then the action ofĜ on C k (R \ Z) is hypercyclic. We also show that the action ofĜ on Cp(ω1) is not hypercyclic.