Complex Dynamics

by Carleson, Lennart; Gamelin, Theodore W.

• ISBN13: 9780387979427
• ISBN10: 0387979425
• Format: Paperback
• Copyright: 1993-06-01
• Publisher: Springer Verlag

Summary

Complex Dynamicsdiscusses the properties of conformal mappings in the complex plane, a subject that is closely connected to the study of fractals and chaos. Indeed the culmination of the book is a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. The book focuses on the analytic side of this contemporary subject. The text was developed out of a course taught over several semesters; its focus is to help students and instructors to familiarize themselves with complex dynamics. Topics covered include: conformal and quasi-conformal mappings, fixed points and conjugations, basic rational iteration (the Julia set), classification of periodic components, critical points and expanding maps, some applications of conformal mappings (e.g. Hermann rings), the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.

Table of Contents

Preface	v
I. Conformal and Quasiconformal Mappings	1	(26)
Some Estimates on Conformal Mappings	1	(4)
The Riemann Mapping	5	(4)
Montel's Theorem	9	(2)
The Hyperbolic Metric	11	(4)
Quasiconformal Mappings	15	(2)
Singular Integral Operators	17	(2)
The Beltrami Equation	19	(8)
II. Fixed Points and Conjugations	27	(26)
Classification of Fixed Points	27	(4)
Attracting Fixed Points	31	(1)
Repelling Fixed Points	32	(1)
Superattracting Fixed Points	33	(2)
Rationally Neutral Fixed Points	35	(6)
Irrationally Neutral Fixed Points	41	(6)
Homeomorphisms of the Circle	47	(6)
III. Basic Rational Iteration	53	(16)
The Julia Set	53	(5)
Counting Cycles	58	(5)
Density of Repelling Periodic Points	63	(2)
Polynomials	65	(4)
IV. Classification of Periodic Components	69	(12)
Sullivan's Theorem	69	(5)
The Classification Theorem	74	(5)
The Wolff-Denjoy Theorem	79	(2)
V. Critical Points and Expanding Maps	81	(18)
Siegel Disks	81	(8)
Hyperbolicity	89	(2)
Subhyperbolicity	91	(2)
Locally Connected Julia Sets	93	(6)
VI. Applications of Quasiconformal Mappings	99	(10)
Polynomial-like Mappings	99	(2)
Quasicircles	101	(2)
Herman Rings	103	(2)
Counting Herman Rings	105	(1)
A Quasiconformal Surgical Procedure	106	(3)
VII. Local Geometry of the Fatou Set	109	(14)
Invariant Spirals	109	(4)
Repelling Arms	113	(4)
John Domains	117	(6)
VIII. Quadratic Polynomials	123	(38)
The Mandelbrot Set	123	(10)
The Hyperbolic Components of M	133	(3)
Green's Function of Jc	136	(3)
Green's Function of M	139	(3)
External Rays with Rational Angles	142	(6)
Misiurewicz Points	148	(5)
Parabolic Points	153	(8)
Epilogue	161	(2)
References	163	(8)
Index	171	(4)
Symbol Index	175