Wandering domains and entire maps of bounded type

Abstract

Complex dynamics is one of the richest and most active branches of dynamical systems. Its goal is to study what happens to analytic functions on the complex plane (or the Riemann sphere) when it is iterated. In this master thesis the focus is on transcendental dynamics since the assumption is that $f:\mathbb{C}\rightarrow \mathbb{C}$ is a transcendental entire function. The foundations of complex dynamics were laid by Pierre Fatou and Gaston Julia in the 1920s when they defined the Fatou and Julia sets, named after them. Roughly speaking, the Fatou set is the stable set since all the points in a neighbourhood have the same behaviour after iteration. Alternatively, the points of the Julia set are those that behave unpredictably after iteration. For that reason the Julia set is also called the chaotic set. Both sets are invariant and give a natural partition of the complex plane. The Fatou set is made up of the complementary domains in $\mathbb{C}$ of the Julia set, the Fatou components. Since it is stable a possible Fatou components. Since it is stable a possible Fatou component $U$ can be either periodic (if $f^p(U)=U$ for some $p \in \mathbb{N})$, pre-periodic (if they are periodic eventually) or wandering (if$f^n(U) \cap f^m(U)=\emptyset$ for $m\neq n$).