A Survey of Baker Wandering Domains

TL;DR

This survey reviews the current understanding of Baker wandering domains in transcendental meromorphic functions, including examples, their influence on dynamics, and classes lacking such domains.

Contribution

It compiles and discusses known examples, properties, and open problems related to Baker wandering domains in complex dynamics.

Findings

First example of Baker wandering domain revisited

Baker wandering domains influence singular values and dynamics

Some classes of functions are shown not to have Baker wandering domains

Abstract

Let $f:\mathbb C\to \widehat{\mathbb C}=\mathbb C \cup\{\infty\}$ be a transcendental meromorphic function (possibly without any pole) with a single essential singularity, and that is chosen to be at $\infty$. The set of points $z\in\mathbb{{\widehat{C}}}$ such that the family of iterates $\{f^n\}_{n\geq 0}$ is defined and forms a normal family in a neighborhood of $z$ is known as the Fatou set of $f$. For a Fatou component $W$, let $W_j$ denote the Fatou component containing $f^j(W)$. A Fatou component $W$ is called wandering if $W_m\bigcap W_n=\emptyset$ for all $m \neq n$. A wandering domain $W$ of $f$ is called a Baker wandering domain, if each $W_n$ is bounded, multiply connected, and $W_n$ surrounds $0$ for all large $n$ and, dist$(W_n,0)\to\infty$ as $n\to\infty$. This paper surveys the current state of knowledge on Baker wandering domains. We revisit the first example of the Baker wandering domain followed by other examples. The influence of Baker wandering domain on the singular values and dynamics of the function is presented. We also discuss some classes of functions that do not possess any Baker wandering domain. Several problems are proposed throughout the article at relevant places.