Dynamics in One Complex Variable. (AM-160): (AM-160) - Third Edition

✍ Scribed by John Milnor

• Publisher: Princeton University Press
• Year: 2011
• Tongue: English
• Leaves: 312
• Series: Annals of Mathematics Studies; 160
• Edition: Third
• 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

Table Of Contents
List of Figures
Preface to the Third Edition
Chronological Table
Riemann Surfaces
Iterated Holomorphic Maps
Local Fixed Point Theory
Periodic Points: Global Theory
Structure of the Fatou Set
Using the Fatou Set to Study the Julia Set
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