The escaping set of transcendental self-maps of the punctured plane

TL;DR

This paper investigates the escape dynamics of points under holomorphic self-maps of the punctured complex plane, revealing diverse escape rates and the complex structure of the fast escaping set.

Contribution

It introduces new methods to construct various escaping orbits and analyzes the structure of the fast escaping set for transcendental self-maps of , highlighting its intricate boundary properties.

Findings

Existence of orbits with different escape rates to 0 and .

Construction of uncountably many disjoint fast escaping sets.

Fast escaping sets have the Julia set as their boundary.

Abstract

We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb C^*=\mathbb C\setminus\{ 0\}$ for which both 0 and $\infty$ are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either 0, $\infty$ or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points each of which has the Julia set as its boundary.