Escaping points of entire functions of small growth

TL;DR

This paper investigates the connectivity of the escaping set of transcendental entire functions with small growth, establishing conditions under which the set is connected and supporting Eremenko's conjecture.

Contribution

It provides new criteria linking the growth order of entire functions to the topological properties of their escaping sets, confirming Eremenko's conjecture in specific cases.

Findings

$I(f)$ is connected for functions with order zero and small growth

$I(f)$ is connected if the function has order less than 1/2 and regular growth

Conditions are given that ensure no unbounded Fatou components exist

Abstract

Let $f$ be a transcendental entire function and let $I(f)$ denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, $I(f)$ is connected. In particular, we show that $I(f)$ is connected if $f$ has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko's conjecture that $I(f)$ has no bounded components is true. We also give a new criterion related to $I(f)$ which is sufficient to ensure that $f$ has no unbounded Fatou components.