On Quasi-nested wandering domains

Abstract

Given a meromorphic function $f$, a wandering domain $W$ is said to be a Quasi-nested wandering domain if there exists a sequence $\{n_k\}$ in $\mathbb{N}$ such that each $U_n$ is bounded, $U_{n_{k}}$ surrounds $0$ for all $k>0$ and $U_{n_{k}}\rightarrow \infty$ as $k\rightarrow \infty$. An omitted value $a\in \widehat{\mathbb{C}}$ is called a \textit{Baker omitted value} of $f$ if for all $r>0$, $f^{-1}(B_{r}(a))=\mathbb{C}\setminus \bigcup^{\infty}_{i=1} D_{i}$ where $D_{i}\cap D_{j}= \phi$ for $i\neq j$ and each $D_{i}$ is a bounded simply connected domain. In this paper,  the nature of the singularity of a particular class $\mathcal{F}$ consisting of all meromorphic functions of the form $f(z)= \frac {1}{h(z)}+a$ for $a\in\mathbb{C}$ and $h$ is an entire function having a Baker wandering domain, lying over the Baker omitted value are discussed. Various dynamical issues of the singular values of $f$ have been studied. Also following are shown in this paper. If $a$  be the Baker omitted value of $f$ then $f$ has a Quasi-nested wandering domain $U$ if and only if there exists $\{n_{k}\}_{k>0}$ such that each $U_{n_{k}}$ surrounds $a$ and $U_{n_{k}}\rightarrow a$ as $k \rightarrow \infty$. If $f$ be a function having Quasi-nested wandering domain then all the Fatou components of $f$ are bounded. In particular, $f$ has no Baker domain. Also existence of Quasi-nested wandering domain ensures that $J_{\infty}$ is a singleton buried component. Finally we have given a result about the non existence of Quasi-nested wandering domain.