Iteration of rational functions : complex analytic dynamical systems / Alan F. BeardonType de document : MonographieCollection : Graduate texts in mathematics, 132Langue : anglais.Pays : Etats Unis.Mention d'édition: reprintingÉditeur : New York : Springer, 2000Description : 1 vol. (XVI-280 p.) : ill. ; 24 cmISBN : 9780387951515.ISSN : 0072-5285.Bibliographie : Bibliogr. p. 273-277. Index.Sujet MSC : 30D05, Entire and meromorphic functions of one complex variable, and related topics, Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30-02, Research exposition (monographs, survey articles) pertaining to functions of a complex variable
37Fxx, Dynamical systems and ergodic theory - Dynamical systems over complex numbers
26C15, Real functions - Polynomials, rational functions in real analysis, Real rational functionsEn-ligne : Zentralblatt | MathSciNet
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The book is the first book completely dedicated to the theory of iteration of rational functions. Here are the titles of its chapters: 1. Examples; 2. Rational maps; 3. The Fatou and Julia sets; 4. Properties of the Julia set; 5. The structure of the Fatou set; 6. Periodic points; 7. Forward invariant components; 8. The no wandering domains theorem; 9. Critical points; 10. Hausdorff dimension; 11. Examples.
Approximately two-thirds of the contents consist of the classical results of Fatou and Julia. Among modern results in this book there are the famous Sullivan no wandering domains theorem with a fairly clear explanation, the Douady and Hubbard theorem that states that the complement of the Mandelbrot set is simply connected, and Shishikura's precise estimate for the number of nonrepelling cycles. The book requires a minimal (say, undergraduate) background in complex analysis and all material is provided with a number of examples and exercises.
In the preface the author writes: "This is not a book for experts, nor is it written by one; it is a modest attempt to lay down the basic foundations of the theory of iteration of rational maps in a clear, precise, complete and rigorous way.'' (MathSciNet)
Bibliogr. p. 273-277. Index