Hypercyclicity of composition operators in Stein manifolds

Abstract

We characterize hypercyclic composition operators C φ : f ↦ f ∘ φ C_\varphi :f\mapsto f\circ \varphi on the space of holomorphic functions on a connected Stein manifold Ω \Omega with φ \varphi being a holomorphic self-map of Ω \Omega . In turns out that in the case when all balls with respect to the Carathéodory pseudodistance are relatively compact in Ω \Omega , a much simpler characterization may be obtained (many natural classes of domains in C N \mathbb {C}^N satisfy this condition). Moreover, we show that in such a class of manifolds, as well as in simply connected and infinitely connected planar domains, hypercyclicity of C φ C_\varphi implies its hereditary hypercyclicity with respect to the full sequence of natural numbers.