Complex Dynamics

by Lennart Carleson, Theodore W. Gamelin

Overview

Complex dynamics is today very much a focus of interest. Though several fine expository articles were available, by P. Blanchard and by M. Yu. Lyubich in particular, until recently there was no single source where students could find the material with proofs. For anyone in our position, gathering and organizing the material required a great deal of work going through preprints and papers and in some cases even finding a proof. We hope that the results of our efforts will be of help to others who plan to learn about complex dynamics and perhaps even lecture. Meanwhile books in the field a. re beginning to appear. The Stony Brook course notes of J. Milnor were particularly welcome and useful. Still we hope that our special emphasis on the analytic side will satisfy a need. This book is a revised and expanded version of notes based on lectures of the first author at UCLA over several \Vinter Quarters, particularly 1986 and 1990. We owe Chris Bishop a great deal of gratitude for supervising the production of course notes, adding new material, and making computer pictures. We have used his computer pictures, and we will also refer to the attractive color graphics in the popular treatment of H. -O. Peitgen and P. Richter. We have benefited from discussions with a number of colleagues, and from suggestions of students in both our courses.

Product Details

ISBN-13:	9780387979427
Publisher:	Springer New York
Publication date:	07/23/1993
Series:	Universitext
Edition description:	1993
Pages:	192
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

I. Conformal and Quasiconformal Mappings.- 1. Some Estimates on Conformal Mappings.- 2. The Riemann Mapping.- 3. Montel’s Theorem.- 4. The Hyperbolic Metric.- 5. Quasiconformal Mappings.- 6. Singular Integral Operators.- 7. The Beltrami Equation.- II. Fixed Points and Conjugations.- 1. Classification of Fixed Points.- 2. Attracting Fixed Points.- 3. Repelling Fixed Points.- 4. Superattracting Fixed Points.- 5. Rationally Neutral Fixed Points.- 6. Irrationally Neutral Fixed Points.- 7. Homeomorphisms of the Circle.- III. Basic Rational Iteration.- 1. The Julia Set.- 2. Counting Cycles.- 3. Density of Repelling Periodic Points.- 4. Polynomials.- IV. Classification of Periodic Components.- 1. Sullivan’s Theorem.- 2. The Classification Theorem.- 3. The Wolff-Denjoy Theorem.- V. Critical Points and Expanding Maps.- 1. Siegel Disks.- 2. Hyperbolicity.- 3. Subhyperbolicity.- 4. Locally Connected Julia Sets.- VI. Applications of Quasiconformal Mappings.- 1. Polynomial-like Mappings.- 2. Quasicircles.- 3. Herman Rings.- 4. Counting Herman Rings.- 5. A Quasiconformal Surgical Procedure.- VII. Local Geometry of the Fatou Set.- 1. Invariant Spirals.- 2. Repelling Arms.- 3. John Domains.- VIII. Quadratic Polynomials.- 1. The Mandelbrot Set.- 2. The Hyperbolic Components of—.- 3. Green’s Function of—c.- 4. Green’s Function of—.- 5. External Rays with Rational Angles.- 6. Misiurewicz Points.- 7. Parabolic Points.- Epilogue.- References.- Symbol Index.

Editorial Reviews

Discusses the behavior of analytic functions under iteration, including conformal and quasiconformal mappings, fixed points and conjugations, the local geometry of the Fatou set, and how Julia sets appear in the Mandelbrot set. For mathematics students with a background in complex analysis. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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