Models for the Eremenko-Lyubich class

Abstract

If $f$ is in the Eremenko-Lyubich class (transcendental entire functions with bounded singular set) then $\hat{I}\text{c◯}=\{z:|f(z)|>R\}$ and $f{|}_{\hat{I}}\text{c◯}$ must satisfy certain simple topological conditions when $R$ is sufficiently large. A model $(\hat{I}\text{c◯},F)$ is an open set $\hat{I}\text{c◯}$ and a holomorphic function $F$ on $\hat{I}\text{c◯}$ 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.

28 pages, 6 figures