Universality of Composition Operators and Applications to Holomorphic Dynamics

TL;DR

This paper explores the universality properties of composition operators induced by transcendental entire functions within complex dynamics, linking linear operator theory with non-linear holomorphic dynamics.

Contribution

It introduces a novel approach to analyze the universality of composition operators using the dynamics of transcendental entire functions on the Fatou set.

Findings

Identifies conditions under which composition operators exhibit universality.

Connects linear operator dynamics with non-linear complex dynamics.

Provides new insights into the structure of Fatou sets and composition operators.

Abstract

By investigating which level of universality composition operators $C_f$ can have, where the symbol $f$ is given by the restriction of a transcendental entire function to suitable parts of the Fatou set of $f$, this work combines the theory of dynamics of continuous linear operators on spaces of holomorphic functions with the theory of non-linear complex dynamics on the complex plane.