On the dynamics of composition of entire functions

Abstract

Let $f$ be an entire function. For $n \in {\bb N}$, let $f^n$ denote the $n$th iterate of $f$. The set \[ F(f)=\{z:(f^n)\hbox{ is normal in some neighbourhood of }z\} \] is the Fatou set or the set of normality and its complement $J(f)$ is the Julia set. If $U$ is a component of $F$$(f)$, then $f$$(U)$ lies in some component $V$ of $F$$(f)$. If $U_n\cap U_m=\phi$ for $n \ne m$ where $U_n$ denotes the component of $F$$(f)$ which contains $f^n(U)$, then $U$ is called a wandering domain, else $U$ is called a pre-periodic domain, and if $U_n = U$ for some $n \in {\bb N}$ then $U$ is called periodic domain.

It is known that for entire functions $f$ and $g$, $f$$(g)$ has wandering domain if and only if $f$$(g)$ has wandering domain. In this paper we discuss the existence of wandering domains of composite entire functions with regards to its factors $f$ and $g$.