On the connectivity of the escaping set in the punctured plane

TL;DR

This paper investigates the connectivity properties of the escaping set in transcendental self-maps of the punctured plane, revealing a trichotomy and constructing examples of doubly connected Baker domains in this setting.

Contribution

It establishes a classification of the escaping set's connectivity in the punctured plane and constructs the first example of doubly connected Baker domains in this context.

Findings

The escaping set is either connected or has infinitely many components.

The set combined with and is either connected or has exactly two components.

Doubly connected Baker domains can exist in , with their closure containing both 0 and .

Abstract

We consider the dynamics of transcendental self-maps of the punctured plane, $\mathbb{C}^*=\mathbb{C}\setminus \{0\}$. We prove that the escaping set $I(f)$ is either connected, or has infinitely many components. We also show that $I(f)\cup \{0,\infty\}$ is either connected, or has exactly two components, one containing $0$ and the other $\infty$. This gives a trichotomy regarding the connectivity of the sets $I(f)$ and $I(f)\cup \{0,\infty\}$, and we give examples of functions for which each case arises. Finally, whereas Baker domains of transcendental entire functions are simply connected, we show that Baker domains can be doubly connected in $\mathbb{C}^*$ by constructing the first such example. We also prove that if $f$ has a doubly connected Baker domain, then its closure contains both $0$ and $\infty$, and hence $I(f)\cup\{0,\infty\}$ is connected.