Dynamics of composition operators on spaces of holomorphic functions on plane domains

TL;DR

This paper investigates the dynamic properties of composition operators on holomorphic function spaces, establishing conditions for hypercyclicity, supercyclicity, and mixing based on domain topology and operator characteristics.

Contribution

It provides a comprehensive characterization of hypercyclic and supercyclic composition operators on various plane domains, linking operator dynamics to domain topology and automorphic properties.

Findings

Any such operator is hypercyclic iff it is topologically mixing.

Supercyclicity for automorphic symbols occurs iff the operator is mixing.

No supercyclic weighted composition operators exist on multiply connected domains with multiple holes.

Abstract

We study the dynamic behaviour of (weighted) composition operators on the space of holomorphic functions on a plane domain. Any such operator is hypercyclic if and only if it is topologically mixing, and when the symbol is automorphic, such an operator is supercyclic if and only if it is mixing. When the domain is a punctured plane, a composition operator is supercyclic if and only if it satisfies the Frequent Hypercyclity Criterion, and when the domain is conformally equivalent to a punctured disc, such an operator is hypercyclic if and only if it satisfies the Frequent Hypercyclicity Criterion. When the domain is finitely connected and either conformally equivalent to an annulus or having two or more holes, no weighted composition operator can be supercyclic.