Hypercyclicity of composition operators in Stein manifolds

TL;DR

This paper characterizes when composition operators are hypercyclic on spaces of holomorphic functions in Stein manifolds, providing simpler criteria under certain geometric conditions and linking hypercyclicity to hereditary hypercyclicity.

Contribution

It offers a new characterization of hypercyclic composition operators on Stein manifolds and establishes conditions under which hypercyclicity implies hereditary hypercyclicity.

Findings

Characterization of hypercyclic composition operators in Stein manifolds.

Simplified criteria for hypercyclicity when Carathéodory balls are relatively compact.

Hypercyclicity implies hereditary hypercyclicity in certain domains.

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.