دانلود کتاب Rational Iteration: Complex Analytic Dynamical Systems

The theory of rational iteration has its origin in long memoires by FATOU and JULIA, based on the KOEBE-POINCARE Uniformization Theorem, MONTEL'S Nor- mality Criterion and earlier work on functional equations due to BÖTTCHER, KCENIGS, LEAU, POINCARE and SCHRÖDER at the turn of the century 1 ·. FATOU and JULIA independently discovered the dichotomy of the RIEMANN sphere into the sets now bearing their names, by considering the sequence of iterates of an arbitrary non-linear rational function. More than sixty years after this fundamental work the field attracted new in- terest in the early eighties, when SULLIVAN announced the solution of the most important problem which had remained open. His no wandering domains theo- rem, on combination with the classification of periodic domains due to FATOU and CREMER, theorems of SIEGEL and ARNOL'D concerning the existence of rotation domains and SHISHIKURA'S precise bound for the number of periodic cycles, yields a rather complete description of the dynamics of a given iteration sequence (/n), that is, of the complex analytic dynamical system (/, C). The empirical discoveries due to MANDELBROT and the beautiful dynamical colour-plates, for example, in the splendidly illustrated volume by PEITGEN and RICHTER, have probably also stimulated new interest in rational iteration. This book is intended to give a self-contained exposition of the theory of FATOU and JULIA and of more recent developments. Apart from some results of general interest being discussed in the first chapter, and some particular topics referred to in places, the only prerequisites are a good knowledge of analytic function theory as may be found in AHLFORS' Complex Analysis. Keeping the student in mind as well as the mathematician who wants to become familiar with the basic theory, I have not made great attempts to present those parts of the theory which are based on quasiconformal mappings. These include part of the work of DOUADY and HUBBARD on polynomial-like mappings and SHISHIKURA'S method of quasiconformal surgery *. Because of its extraordinary importance, however, I have included a proof of SULLIVAN'S Theorem, but am conscious that the proof remains unsatisfactory, lack of space having precluded a rigorous presentation of its foundations. Nevertheless, the book may serve as a textbook for a course following a one-year introduction to analytic function theory, and should prove useful for the advanced student as well as the mathematician who wants to become acquainted with this field. However, I do hope that the research worker will also find some aspects new. Many of the results appear for the first time in book form, and indeed some seem to have never been published before. This applies also, as far as I am aware, to some of the proofs of known results. The book is divided into six chapters which are subdivided into sections. Each section is provided with a list of exercises. Most of them are purely mathematical exercises ("Prove that...", and the reader is urgently requested to do this), but also exercises which should stimulate the reader to do experimental mathematics. The figures created by Turbo Pascal programs serve to illustrate various theorems and phenomena (some of the figures have been rotated; the values of parameters are rounded). While the contents of Chapters 2, 3 and 4 are more or less canonical, some of the selected material in Chapters 5 and 6 reflects my own interests.