Hypercyclicity of composition operators in Stein manifolds

Abstract

We characterise hypercyclic composition operators $C_\varphi:f\mapsto f\circ\varphi$ on the space of functions holomorphic on $\Omega$, where $\Omega$ is a connected Stein manifold and $\varphi$ is a holomorphic self-mapping of $\Omega$. In the case when all balls with respect to the Carath\'{e}odory pseudodistance are relatively compact in $\Omega$, 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 $C_\varphi$ implies its hereditary hypercyclicity.