Iteration of rational functions : complex analytic dynamical systems有理函数的重复
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作者： Alan F. Beardon著
出 版 社：
出版时间： 2000-9-1字数：版次： 1页数： 280印刷时间： 2000/09/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387951515包装： 平装内容简介
This book makes available a comprehensive, detailed, and organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The material covered extends from the original memoirs of Fatou and Julia to the recent and important results and methods of Sullivan and Shishikura. Many of the details of the proofs have not occurred in print before. The theory of of dynamical systems and chaos has recently undergone a rapid growth in popularity, in part due to the spectacular computer graphics of Julia sets, fractals, and the Mandelbrot set. This text focuses on the specialized area of complex analytic dynamics, a subject that dates back to 1916 and is currently a very active area in mathematics.
目录
Preface
Prerequisites
CHAPTER 1
Examples
1.1. Introduction
1.2. Iteration of M6bius Transformations
1.3. Iteration of z—z2—1
1.4. Tchebychev Polynomials
1.5. Iteration of z—z2—1
1.6. Iteration ofz—zz＋c
1.7. Iteration ofz—z＋1/z
1.8. Iteration ofz—2z—1/z
1.9. Newton's Approximation
1.10. General Remarks
CHAPTER 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
CHAPTER 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
CHAPTER 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
CHAPTER 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
CHAPTER 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
CHAPTER 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 oflnvariant Components
CHAPTER 8 The No Wandering Domains Thorem
CHAPTER 9 Critical Points
CHAPTER 10 Hausdorff Dimension
CHAPTER 11 Examples
References
Index of Examples
Index
