Fast escaping points of entire functions

TL;DR

This paper investigates the structure of the fastest escaping points in transcendental entire functions, revealing new properties of their level sets, boundary behavior, and connections to longstanding conjectures in complex dynamics.

Contribution

It introduces a novel level-based decomposition of the escaping set and establishes new results on its structure and boundary, linking Baker's and Eremenko's conjectures.

Findings

The escaping set can be expressed as a union of closed levels with a specific structure.

Boundaries of Fatou components in the escaping set are contained within the set itself.

Many functions have an escaping set and levels forming an infinite spider's web, with strong dynamical properties.

Abstract

Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$, we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$, then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$. In particular, we study functions for which $A(f)$, and each of its levels, has the structure of an `infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.