Entire functions arising from trees

Abstract

Given any infinite tree in the plane satisfying certain topological conditions, we construct an entire function $f$ with only two critical values $\pm 1$ and no asymptotic values such that $f^{-1}([-1,1])$ is ambiently homeomorphic to the given tree. This can be viewed as a generalization of a result of Grothendieck to the case of infinite trees. Moreover, a similar idea leads to a new proof of a result of Nevanlinna and Elfving.

22 pages, 16 figures

References in corpus (1)

• True trees are dense

Cited by in corpus (1)

• Hausdorff dimension of escaping sets of meromorphic functions