Lebesgue measure of escaping sets of entire functions

TL;DR

This paper investigates conditions under which the escaping sets of certain entire functions have zero Lebesgue measure, extending previous results and constructing examples to demonstrate sharpness of these conditions.

Contribution

It provides new criteria for zero Lebesgue measure of escaping sets in the Eremenko-Lyubich class and extends results to infinite order entire functions.

Findings

Conditions for zero Lebesgue measure of escaping sets in class $\

,

constructed an entire function showing sharpness of previous conditions,

Abstract

For a transcendental entire function $f$ of finite order in the Eremenko-Lyubich class $\mathcal{B}$, we give conditions under which the Lebesgue measure of the escaping set $\mathcal{I}(f)$ of $f$ is zero. This is inspired by the recent work of Aspenberg and Bergweiler, in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko-Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to the case of infinite order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.