Interactions between Universal Composition Operators and Complex Dynamics

TL;DR

This paper explores the universality properties of composition operators induced by transcendental entire functions on specific Fatou set components, revealing new insights into their behavior near periodic points and infinity.

Contribution

It establishes the universality of composition operators on Baker and wandering domains and extends results to weighted composition operators in complex dynamics.

Findings

Universality of $C_f$ on Baker and wandering domains.

Behavior of universal vectors near periodic points and infinity.

Universality results for weighted composition operators in Fatou components.

Abstract

This paper is concerned with universality properties of composition operators $C_f$, where the symbol $f$ is given by a transcendental entire function restricted to parts of its Fatou set. We determine universality of $C_f$ when $f$ is restricted to (subsets of) Baker and wandering domains. We then describe the behaviour of universal vectors, under the action of iterates of the symbol $f$, near periodic points of $f$ or near infinity. Finally, we establish a principal universality theorem for the more general class of weighted composition operators, which we then apply to uncover universality results in the context of various types of Fatou components of the associated symbol.