Universal sequences of composition operators

TL;DR

This paper characterizes when sequences of holomorphic maps induce dense orbits of functions under composition, extending previous results and linking domain topology with universality properties of holomorphic functions.

Contribution

It provides necessary and sufficient conditions for the existence of dense orbits under composition operators between planar domains, generalizing prior work to different domain pairs.

Findings

Established conditions for dense orbits in $H(G)$ under composition

Extended Grosse-Erdmann and Mortini's results to different domains

Linked topological properties of domains with universality phenomena

Abstract

Let $G$ and $\Omega$ be two planar domains. We give necessary and sufficient conditions on a sequence $(\phi_n)$ of eventually injective holomorphic mappings from $G$ to $\Omega$ for the existence of a function $f\in H(\Omega)$ whose orbit under the composition by $(\phi_n)$ is dense in $H(G)$. This extends a result of the same nature obtained by Grosse-Erdmann and Mortini when $G=\Omega$. An interconnexion between the topological properties of $G$ and $\Omega$ appears. Further, in order to exhibit in a natural way holomorphic functions with wild boundary behaviour on planar domains, we study a certain type of universality for sequences of continuous mappings from a union of Jordan curves to a domain.