Models for the Speiser class

Abstract

The Eremenko–Lyubich class B consists of transcendental entire functions with bounded singular set and the Speiser class S ⊂ B is made up of functions with a finite singular set. In an earlier work (J. Lond. Math. Soc. 92 (2015) 202–221), I gave a method for constructing Eremenko–Lyubich functions that approximate certain simpler functions called models. In this paper, I show that all models can be approximated in a weaker sense by Speiser class functions, and that the stronger approximation of the earlier work (J. Lond. Math. Soc. 92 (2015) 202–221) 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.