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Viser: The Iteration of Rational Functions - Complex Analytic Dynamical Systems
The Iteration of Rational Functions
Complex Analytic Dynamical Systems
Alan F. Beardon
(2000)
Sprog: Engelsk
om ca. 10 hverdage
Detaljer om varen
- 1. Udgave
- Paperback
- Udgiver: Springer New York (September 2000)
- ISBN: 9780387951515
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. LimitFunctions.-
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.