Entire functions arising from trees

TL;DR

This paper constructs entire functions from infinite trees with specific properties, generalizing Grothendieck's result and providing a new proof of classical theorems in complex analysis.

Contribution

It introduces a method to associate entire functions with infinite trees, extending existing results and offering new proofs of known theorems.

Findings

Constructed entire functions with two critical values from infinite trees

Generalized Grothendieck's result to infinite trees

Provided a new proof of Nevanlinna and Elfving's theorem

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.