Models for the Speiser class

TL;DR

This paper explores the relationship between Speiser and Eremenko-Lyubich classes of transcendental entire functions, showing how models for the former can be approximated by functions in the latter and highlighting geometric differences.

Contribution

It demonstrates that Speiser class functions can approximate models of Eremenko-Lyubich functions in a weaker sense and identifies geometric restrictions unique to Speiser functions.

Findings

Models for Speiser class can be approximated by Eremenko-Lyubich functions in a weaker sense.

Stronger approximation methods for Eremenko-Lyubich functions may not apply to Speiser class functions.

Geometric restrictions are identified that distinguish Speiser class functions from general Eremenko-Lyubich functions.

Abstract

The Eremenko-Lyubich class consists of transcendental entire functions with bounded singular set and the Speiser class is made up of functions with a finite singular set. In an earlier paper "Models for the Eremenko-Lyubich class" I gave a method for constructing Eremenko-Lyubich functions that approximate certain simpler functions called models. In this paper, I show that all such models can be approximated in a weaker sense by Speiser class functions, and that the stronger approximation possible using Eemenko-Lyubich functions can fail for the Speiser class. In particular, I give geometric restrictions on the geometry of a Speiser class function that need not be satisfied by general Eremenko-Lyubich functions.