Iteration of Rational Functions: Complex Analytic Dynamical Systems

1 Examples.- 1.1. Introduction.- 1.2. Iteration of Möbius 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.