On invariance of order and the area property for finite-type entire functions

TL;DR

This paper investigates whether the growth order of finite-type entire functions is determined by their combinatorial structure, exploring the area property and its implications in conformal dynamics and function theory.

Contribution

It introduces the area property as a key condition linking combinatorial data to the order of entire functions, and discusses its relevance to open conjectures and dynamics.

Findings

The order of entire functions may not be invariant under topological equivalence.

The area property is connected to important conjectures in conformal dynamics.

Evidence suggests invariance of order and the area property do not hold universally.

Abstract

Let f be an entire function that has only finitely many critical and asymptotic values. Up to topological equivalence, the function $f$ is determined by combinatorial information, more precisely by an infinite graph known as a "line-complex". In this note, we discuss the natural question whether the order of growth of an entire function is determined by this combinatorial information. The search for conditions that imply a positive answer to this question leads us to the "area property", which turns out to be related to many interesting and important questions in conformal dynamics and function theory. These include a conjecture of Eremenko and Lyubich, the measurable dynamics of entire functions, and pushforwards of quadratic differentials. We also discuss evidence that invariance of order and the area property fail in general.