Approximation Theory has been one of the most important tools for constructing functions in Complex Dynamics. In this mini-course, we will give an overview of results and techniques from Approximation Theory that have been used in the area, including Runge’s theorem and Arakelyan's theorem. We will present several versions of these results which help us construct transcendental entire and meromorphic functions with specific properties. We will also give an overview of a paper by Eremenko and Lyubich, which includes some of the earliest applications of Approximation Theory in Transcendental Dynamics.

In the past few years, there has been a resurge in the use of Approximation Theory to obtain wandering domains with interesting dynamics and topology. Benini, Evdoridou, Fagella, Rippon and Stallard constructed examples of wandering domains with several types of internal dynamics that were unknown before. More recently, Marti-Pete, Rempe and Waterman, inspired by work of Boc Thaler, constructed wandering domains with interesting topology, including wandering domains that form Lakes of Wada. We will discuss these two constructions in detail and present some open questions in this area.

Let U be a Fatou component of a transcendental entire function. If U is not eventually periodic then it is called a wandering domain. Although Sullivan's celebrated result showed that rational maps have no wandering domains, transcendental entire functions can have wandering domains. In this talk we will look at internal dynamics of simply connected wandering domains and we will give a nine-way classification of these domains. We will also show that all nine types of wandering domains that arise from this classification are realizable. This is joint work with Anna Miriam Benini, Nuria Fagella, Phil Rippon and Gwyneth Stallard.