Escaping Fatou components of transcendental self-maps of the punctured plane

TL;DR

This paper constructs explicit examples of transcendental self-maps of the punctured plane with escaping Fatou components, including wandering and Baker domains, advancing understanding of complex dynamics in this setting.

Contribution

It introduces the first explicit examples of such maps with escaping Fatou components and develops conditions for their existence.

Findings

Constructed functions with escaping Fatou components

Provided explicit examples with Baker and wandering domains

Developed a sufficient condition for simply connected escaping wandering domains

Abstract

We study the iteration of transcendental self-maps of $\mathbb{C}^*:=\mathbb{C}\setminus \{0\}$, that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.