Hypercyclic and topologically transitive semigroups of composition operators (revised and extended version)

Abstract

Let $X$ be a separable locally compact metric space, let $E\neq\{0\}$ be a normed space, and let $G$ be a semigroup of continuous self-maps of $X$. Every $φ\in G$ induces the composition operator $C_φf=f\circφ$ on the space $C(X,E)$ of continuous functions endowed with the compact-open topology. We characterize the topological transitivity of the semigroup $\widehat{G}=\{C_φ:φ\in G\}$ by a condition that combines the run-away property of $G$ with an approximate form of injectivity, and we show that, when $E$ is separable, $\widehat{G}$ is topologically transitive if and only if it is hypercyclic. In particular, if every element of $G$ is one-to-one and $G$ is run-away, then $\widehat{G}$ is topologically transitive, and hypercyclic when $E$ is separable. For a single map $φ$ we obtain: $C_φ$ is hypercyclic $\iff$ $C_φ$ is topologically transitive $\iff$ $φ$ is one-to-one and run-away, and $C_φ$ is mixing $\iff$ $φ$ is one-to-one and strongly run-away. If, moreover, $φ$ is an open map (which is automatic on manifolds by invariance of domain), then $C_φ$ is Devaney chaotic. On an open interval all of these properties are equivalent to $φ$ being one-to-one without fixed points.