A Survey of 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.