Hypercyclicity of composition operators in Stein manifolds

Abstract

We characterise hypercyclic composition operators $C_φ:f\mapsto f\circφ$ on the space of functions holomorphic on $Ω$, where $Ω$ is a connected Stein manifold and $φ$ is a holomorphic self-mapping of $Ω$. In the case when all balls with respect to the Carathéodory pseudodistance are relatively compact in $Ω$, we show that much simpler characterisation is possible (many natural classes of domains in $\CC^N$ satisfy this condition). Moreover, we show that in such a class of manifolds, and in simply connected and infinitely connected planar domains, hypercyclicity of $C_φ$ implies its hereditary hypercyclicity.