Dynamics in One Complex Variable. (AM-160) : (AM-160).
Title:
Dynamics in One Complex Variable. (AM-160) : (AM-160).
Author:
Milnor, John.
ISBN:
9781400835539
Personal Author:
Edition:
3rd ed.
Physical Description:
1 online resource (313 pages)
Series:
Annals of Mathematics Studies ; v.160
Annals of Mathematics Studies
Contents:
Cover -- Title -- Copyright -- Table of Contents -- List of Figures -- Preface to the Third Edition -- Chronological Table -- Riemann Surfaces -- 1. Simply Connected Surfaces -- 2. Universal Coverings and the Poincaré Metric -- 3. Normal Families: Montel's Theorem -- Iterated Holomorphic Maps -- 4. Fatou and Julia: Dynamics on the Riemann Sphere -- 5. Dynamics on Hyperbolic Surfaces -- 6. Dynamics on Euclidean Surface -- 7. Smooth Julia Sets -- Local Fixed Point Theory -- 8. 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 Theory -- 12. The Holomorphic Fixed Point Formula -- 13. Most Periodic Orbits Repel -- 14. Repelling Cycles Are Dense in J -- Structure of the Fatou Set -- 15. Herman Rings -- 16. The Sullivan Classification of Fatou Components -- Using the Fatou Set to Study the Julia Set -- 17. Prime Ends and Local Connectivity -- 18. Polynomial Dynamics: External Rays -- 19. Hyperbolic and Subhyperbolic Maps -- Appendices -- 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 -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Y -- Z.
Abstract:
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 a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
Genre:
Electronic Access:
Click to View