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

Dynamics in One Complex Variable. (AM-160)

(AM-160) - Third Edition

By: J. Milnor


Published: 1st January 2006
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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 Lattes map has been made more inclusive, and the Ecalle-Voronin theory of parabolic points is described. The residu iteratif 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.

"John Milnor's book provides a solid foundation and the kind of bird's eye view that perhaps only a mathematician of his caliber can offer."--William J. Satzer, MAA Reviews

List of Figuresp. vi
Preface to the Third Editionp. vii
Chronological Tablep. viii
Riemann Surfaces
Simply Connected Surfacesp. 1
Universal Coverings and the Poincare Metricp. 13
Normal Families: Montel's Theoremp. 30
Iterated Holomorphic Maps
Fatou and Julia: Dynamics on the Riemann Spherep. 39
Dynamics on Hyperbolic Surfacesp. 56
Dynamics on Euclidean Surfacesp. 65
Smooth Julia Setsp. 69
Local Fixed Point Theory
Geometrically Attracting or Repelling Fixed Pointsp. 76
Bottcher's Theorem and Polynomial Dynamicsp. 90
Parabolic Fixed Points: The Leau-Fatou Flowerp. 104
Cremer Points and Siegel Disksp. 125
Periodic Points: Global Theory
The Holomorphic Fixed Point Formulap. 142
Most Periodic Orbits Repelp. 153
Repelling Cycles Are Dense in Jp. 156
Structure of the Fatou Set
Herman Ringsp. 161
The Sullivan Classification of Fatou Componentsp. 167
Using the Fatou Set to Study the Julia Set
Prime Ends and Local Connectivityp. 174
Polynomial Dynamics: External Raysp. 188
Hyperbolic and Subhyperbolic Mapsp. 205
Theorems from Classical Analysisp. 219
Length-Area-Modulus Inequalitiesp. 226
Rotations, Continued Fractions, and Rational Approximationp. 234
Two or More Complex Variablesp. 246
Branched Coverings and Orbifoldsp. 254
No Wandering Fatou Componentsp. 259
Parameter Spacesp. 266
Computer Graphics and Effective Computationp. 271
Referencesp. 277
Indexp. 293
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691124889
ISBN-10: 0691124884
Series: Annals of Mathematics Studies
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 320
Published: 1st January 2006
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 25.4 x 17.8  x 1.91
Weight (kg): 0.56
Edition Number: 3
Edition Type: Revised