Dynamics in One Complex Variable

✍ Scribed by John Milnor

• Publisher: Princeton University Press
• Year: 2006
• Tongue: English
• Leaves: 313
• Series: Annals of Mathematics Studies 160
• Edition: 3
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover......Page 1
Title......Page 4
Copyright......Page 5
Table of Contents......Page 6
List of Figures......Page 7
Preface to the Third Edition......Page 8
Chronological Table......Page 9
1. Simply Connected Surfaces......Page 10
2. Universal Coverings and the Poincaré Metric......Page 22
3. Normal Families: Montel's Theorem......Page 39
4. Fatou and Julia: Dynamics on the Riemann Sphere......Page 48
5. Dynamics on Hyperbolic Surfaces......Page 65
6. Dynamics on Euclidean Surface......Page 74
7. Smooth Julia Sets......Page 78
8. Geometrically Attracting or Repelling Fixed Points......Page 85
9. Böttcher's Theorem and Polynomial Dynamics......Page 99
10. Parabolic Fixed Points: The Leau–Fatou Flower......Page 113
11. Cremer Points and Siegel Disks......Page 134
12. The Holomorphic Fixed Point Formula......Page 151
13. Most Periodic Orbits Repel......Page 162
14. Repelling Cycles Are Dense in J......Page 165
15. Herman Rings......Page 170
16. The Sullivan Classification of Fatou Components......Page 176
17. Prime Ends and Local Connectivity......Page 183
18. Polynomial Dynamics: External Rays......Page 197
19. Hyperbolic and Subhyperbolic Maps......Page 214
Appendix A. Theorems from Classical Analysis......Page 228
Appendix B. Length-Area-Modulus Inequalities......Page 235
Appendix C. Rotations, Continued Fractions, and Rational Approximation......Page 243
Appendix D. Two or More Complex Variables......Page 255
Appendix E. Branched Coverings and Orbifolds......Page 263
Appendix F. No Wandering Fatou Components......Page 268
Appendix G. Parameter Spaces......Page 275
Appendix H. Computer Graphics and Effective Computation......Page 280
References......Page 286
B......Page 302
C......Page 303
D......Page 304
F......Page 305
H......Page 306
J......Page 307
M......Page 308
O......Page 309
P......Page 310
R......Page 311
T......Page 312
Z......Page 313