Classification of periodic Fatou components for rational maps

Abstract

[en] The dynamics of rational maps on the Riemann sphere is one of the most beautiful and well-known topics in complex dynamics. As holomorphic functions, they can be treated both as power series from the analytic point of view, and as locally conformal mappings from the geometric one. Therefore all powerful tools from complex analysis can be used, gathering the work of many analysts, geometers, topologists and algebraists over the time. Its origins date back to the manuscripts of Pierre Fatou and Gaston Julia submitted independently to the 1918 Grand Prize in Mathematics promoted by the French Academy of Sciences. Both of them were inspired a few months before by the theory of normal families of the also French mathematician Paul Montel. They realized that sequences of functions in Montel’s work could be treated as iterates of a certain map, hence the complex plane could be partitioned into a normality and non-normality sets, nowadays known as Fatou and Julia sets, respectively, with stunning topology and dynamics. Actually the Julia set corresponds to those points with a chaotic behaviour after many iterations, and the Fatou set, as its complementary, can be seen as the set of stable or well-behaved points. Fatou focused his attention on what he called singular domains, since he believed in the existence of components of the normality set that were not in a domain of attraction for a periodic cycle, although it was always wrongly refused by Julia. A few years later Cremer noticed that these hypothetical domains for rational maps should be doubly connected, anticipating Herman rings. In fact he pointed out that the local behaviour near periodic points was not always clear. Later on, many important mathematicians such as Siegel, Arnol’d, Herman or Baker contributed on the topic, but there were many unanswered questions left and gradually the subject lost its interest during the next sixty years. Fortunately in the early 1980s powerful computer graphics of complex dynamical systems, as well as the fractal geometry and its applications introduced by Mandelbrot, revitalised the study of rational maps. There was a strong interest in the final fate or long-term behaviour of the points on the Fatou and Julia sets, as well as in the properties of their components. At the same time one of the greatest and the most revolutionary contributions in the complex dynamics field was brought to light by Sullivan’s No-wandering domains Theorem, a wonderful solution to the old Fatou’s conjecture using the recent quasiconformal surgery theory. This result was pleasantly enhanced by the Classification Theorem of periodic domains already developed by Fatou and Cremer, and by some theorems provided by Siegel, Arnol’d and Herman on the existence of rotation domains. This was complemented recently with the Shishikura’s sharp inequality for the number of periodic cycles in the Fatou set proved thanks to the relation between critical points and Fatou components, using quasiconformal surgery for his Master’s thesis in 1987. All these important results, which were developed throughout the last century, configure a very complete and simple description of the dynamics of rational maps. The central topic of the current work is the Classification of periodic Fatou components for a rational map into the five possibilities: attractive basins, superattractive basins, parabolic basins, Siegel discs and Herman rings. Several versions of this central theorem have appeared [7, 9, 12] using alternative echniques. Our intention is to give a clear, precise and self-contained proof of this theorem, based on key tools from complex analysis, iteration theory and hyperbolic geometry. For this purpose, these topics are presented in a coherent and reasonable manner. We have made an effort to show the proofs in a way as simple as possible by gathering geometric and analytic ideas in the literature. Important information and detailed exposition of theses subjects as well as further topics in complex dynamics are included in the books of Milnor [9], Beardon [2], Steinmetz [12] or Keen and Lakicv [6], for example.