Iteration of rational functions : complex analytic dynamical systems

Alan F. Beardon

This book focuses on complex analytic dynamics, which dates from 1916 and is currently attracting considerable interest. The text provides a comprehensive, well-organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The coverage extends from early memoirs of Fatou and Julia to important recent results and methods of Sullivan and Shishikura. Many details of the proofs have not appeared in print before.

「Nielsen BookData」より

[目次]

  • 1 Examples.- 1.1. Introduction.- 1.2. Iteration of Mobius Transformations.- 1.3. Iteration of z ? z2.- 1.4. Tchebychev Polynomials.- 1.5. Iteration of z ? z2 ? 1.- 1.6. Iteration of z ? z2 + c.- 1.7. Iteration of z ? z + 1/z.- 1.8. Iteration of z ? 2z ? 1/z.- 1.9. Newton's Approximation.- 1.10. General Remarks.- 2 Rational Maps.- 2.1. The Extended Complex Plane.- 2.2. Rational Maps.- 2.3. The Lipschitz Condition.- 2.4. Conjugacy.- 2.5. Valency.- 2.6. Fixed Points.- 2.7. Critical Points.- 2.8. A Topology on the Rational Functions.- 3 The Fatou and Julia Sets.- 3.1. The Fatou and Julia Sets.- 3.2. Completely Invariant Sets.- 3.3. Normal Families and Equicontinuity.- Appendix I. The Hyperbolic Metric.- 4 Properties of the Julia Set.- 4.1. Exceptional Points.- 4.2. Properties of the Julia Set.- 4.3. Rational Maps with Empty Fatou Set.- Appendix II. Elliptic Functions.- 5 The Structure of the Fatou Set.- 5.1. The Topology of the Sphere.- 5.2. Completely Invariant Components of the Fatou Set.- 5.3. The Euler Characteristic.- 5.4. The Riemann-Hurwitz Formula for Covering Maps.- 5.5. Maps Between Components of the Fatou Set.- 5.6. The Number of Components of the Fatou Set.- 5.7. Components of the Julia Set.- 6 Periodic Points.- 6.1. The Classification of Periodic Points.- 6.2. The Existence of Periodic Points.- 6.3. (Super) Attracting Cycles.- 6.4. Repelling Cycles.- 6.5. Rationally Indifferent Cycles.- 6.6. Irrationally Indifferent Cycles in F.- 6.7. Irrationally Indifferent Cycles in J.- 6.8. The Proof of the Existence of Periodic Points.- 6.9. The Julia Set and Periodic Points.- 6.10. Local Conjugacy.- Appendix III. Infinite Products.- Appendix IV. The Universal Covering Surface.- 7 Forward Invariant Components.- 7.1. The Five Possibilities.- 7.2. Limit Functions.- 7.3. Parabolic Domains.- 7.4. Siegel Discs and Herman Rings.- 7.5. Connectivity of Invariant Components.- 8 The No Wandering Domains Theorem.- 8.1. The No Wandering Domains Theorem.- 8.2. A Preliminary Result.- 8.3. Conformal Structures.- 8.4. Quasiconformal Conjugates of Rational Maps.- 8.5. Boundary Values of Conjugate Maps.- 8.6. The Proof of Theorem 8.1.2.- 9 Critical Points.- 9.1. Introductory Remarks.- 9.2. The Normality of Inverse Maps.- 9.3. Critical Points and Periodic Domains.- 9.4. Applications.- 9.5. The Fatou Set of a Polynomial.- 9.6. The Number of Non-Repelling Cycles.- 9.7. Expanding Maps.- 9.8. Julia Sets as Cantor Sets.- 9.9. Julia Sets as Jordan Curves.- 9.10. The Mandelbrot Set.- 10 Hausdorff Dimension.- 10.1. Hausdorff Dimension.- 10.2. Computing Dimensions.- 10.3. The Dimension of Julia Sets.- 11 Examples.- 11.1. Smooth Julia Sets.- 11.2. Dendrites.- 11.3. Components of F of Infinite Connectivity.- 11.4. F with Infinitely Connected and Simply Connected Components.- 11.5. J with Infinitely Many Non-Degenerate Components.- 11.6. F of Infinite Connectivity with Critical Points in J.- 11.7. A Finitely Connected Component of F.- 11.8. J Is a Cantor Set of Circles.- 11.9. The Function (z ? 2)2/z2.- References.- Index of Examples.

「Nielsen BookData」より

この本の情報

書名 Iteration of rational functions : complex analytic dynamical systems
著作者等 Beardon, Alan F.
Beardon A.F.
シリーズ名 Graduate texts in mathematics
出版元 Springer-Verlag
刊行年月 2000, c1991
版表示 Softcover reprint of the original 1st ed. 1991
ページ数 xvi, 280 p.
大きさ 24 cm
ISBN 0387951512
NCID BA51187671
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国
この本を: 
このエントリーをはてなブックマークに追加

このページを印刷

外部サイトで検索

この本と繋がる本を検索

ウィキペディアから連想