Classification and Structure of Periodic Fatou Components

For a given rational map f : Ĉ→ Ĉ, the Julia set consists of those points in Ĉ around which the dynamics of the map is chaotic (a notion that can be defined rigorously), while the Fatou set is defined as the complement. The Fatou set, where the dynamics is well-behaved, is an open set, and one can classify its periodic connected components into five well-understood categories. This classification theorem is the focus of the paper, and we attempt to present its proof in an efficient, self-contained, and wellmotivated manner. The proof makes heavy use of methods of hyperbolic geometry on certain open subsets of Ĉ. We develop the theory needed to carry out this analysis. One fundamental result that is often used is the Uniformization Theorem, whose proof in the general case would take us far afield from the usual subject matter of complex dynamics. We present a simpler proof for the case of plane domains, which is all that is needed for the Fatou component classification theorem. Finally we show that each of the five types of Fatou components actually occurs, and we present some of the theory associated with the structure of each.