1 - Introduction

Summary

It is fair to say that the subject known today as complex dynamics – the study of iterations of analytic functions – originated in the pioneering works of P. Fatou and G. Julia early in the twentieth century (see the references [Fat] and [Ju]). In possession of what was then a new tool, Montel's theorem on normal families, Fatou and Julia each investigated the iteration of rational maps of the Riemann sphere and found that these dynamical systems had an extremely rich orbit structure. They observed that each rational map produced a dichotomy of behavior for points on the Riemann sphere. Some points – constituting a totally invariant open set known today as the Fatou set – showed an essentially dissipative or wandering character under iteration by the map. The remaining points formed a totally invariant compact set, today called the Julia set. The dynamics of a rational map on its Julia set showed a very complicated recurrent behavior, with transitive orbits and a dense subset of periodic points. Since the Julia set seemed so difficult to analyse, Fatou turned his attention to its complement (the Fatou set). The components of the Fatou set are mapped to other components, and Fatou observed that these seemed to eventually to fall into a periodic cycle of components. Unable to prove this fact, but able to verify it for many examples, Fatou nevertheless conjectured that rational maps have no wandering domains.