Hypercyclic and topologically transitive semigroups of composition operators

We study the dynamical properties of semigroups of composition operators of continuous functions on a separable locally compact metric space X. In particular, we show that if E is a normed vector space and ϕ : X → X is one-to-one and run-away on X, then the composition operator f ↦ ϕ ∘ f on C ( X , E ) is topologically transitive and hypercyclic, where C ( X , E ) is the space of continuous functions from X into E with the compact-open topology.