Dynamics in one complex variable
This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form...
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| Main Author: | |
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| Format: | eBook Book |
| Language: | English |
| Published: |
Princeton, N.J
Princeton University Press
2006
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| Edition: | 3rd ed. |
| Series: | Annals of mathematics studies |
| Subjects: | |
| ISBN: | 0691124884, 1400835534, 9780691124889, 9781400835539, 9780691124872, 0691124876 |
| Online Access: | Get full text |
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Table of Contents:
- Dynamics in one complex variable -- Table of Contents -- List of Figures -- Preface to the Third Edition -- Chronological Table -- Riemann Surfaces1. Simply Connected Surfaces -- 2. Universal Coverings and the Poincaré Metric -- 3. Normal Families: Montel's Theorem -- Iterated Holomorphic Maps4. Fatou and Julia: Dynamics on the Riemann Sphere -- 5. Dynamics on Hyperbolic Surfaces -- 6. Dynamics on Euclidean Surfaces -- 7. Smooth Julia Sets -- Local Fixed Point Theory8. Geometrically Attracting or Repelling Fixed Points -- 9. Böttcher's Theorem and Polynomial Dynamics -- 10. Parabolic Fixed Points: The Leau-Fatou Flower -- 11. Cremer Points and Siegel Disks -- Periodic Points: Global Theory12. The Holomorphic Fixed Point Formula -- 13. Most Periodic Orbits Repel -- 14. Repelling Cycles are Dense in J -- Structure of the Fatou Set15. Herman Rings -- 16. The Sullivan Classification of Fatou Components -- Using the Fatou Set to Study the Julia Set17. Prime Ends and Local Connectivity -- 18. Polynomial Dynamics: External Rays -- 19. Hyperbolic and Subhyperbolic Maps -- Appendix A: Theorems from Classical Analysis -- Appendix B: Length-Area-Modulus Inequalities -- Appendix C: Rotations, Continued Fractions, and Rational Approximation -- Appendix D: Two or More Complex Variables -- Appendix E: Branched Coverings and Orbifolds -- Appendix F: No Wandering Fatou Components -- Appendix G: Parameter Spaces -- Appendix H: Computer Graphics and Effective Computation -- References -- Index.