| 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 |
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