Models for the Eremenko-Lyubich class

TL;DR

This paper introduces a method to approximate models of the Eremenko-Lyubich class, enabling easier construction of functions with desired properties within this class by focusing on simpler topological models.

Contribution

It establishes that any model satisfying certain conditions can be approximated by an Eremenko-Lyubich function, simplifying the process of constructing functions with specific properties.

Findings

Any model can be approximated by an Eremenko-Lyubich function

Construction of functions with desired properties reduces to building suitable models

Facilitates easier creation of functions with specific dynamical properties

Abstract

If $f$ is in the Eremenko-Lyubich class (transcendental entire functions with bounded singular set) then $\Omega= \{ z: |f(z)| > R\}$ and $f|_\Omega$ must satisfy certain simple topological conditions when $R$ is sufficiently large. A model $(\Omega, F)$ is an open set $\Omega$ and a holomorphic function $F$ on $\Omega$ that satisfy these same conditions. We show that any model can be approximated by an Eremenko-Lyubich function in a precise sense. In many cases, this allows the construction of functions in the Eremenko-Lyubich with a desired property to be reduced to the construction of a model with that property, and this is often much easier to do.