Models for the Eremenko-Lyubich class

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