A discussion of the properties of conformal mappings in the complex plane, closely related to the study of fractals and chaos. Indeed, the book ends in a detailed study of the famous Mandelbrot set, which describes very general properties of such mappings. Focusing on the analytic side of this contemporary subject, the text was developed from a course taught over several semesters and aims 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, classification of periodic components, critical points and expanding maps, some applications of conformal mappings, the local geometry of the Fatou set, and quadratic polynomials and the Mandelbrot set.
Author(s): Lennart Carleson, Theodore W. Gamelin
Series: Universitext
Edition: Corrected
Publisher: Springer
Year: 1993
Language: English
Pages: 186
Tags: Математика;Нелинейная динамика;
Cover......Page 1
Title: Complex Dynamics......Page 3
ISBN 3-540-7942-5......Page 5
Preface......Page 7
Contents......Page 9
1. Some Estimates on Conformal Mappings......Page 12
2. The Riemann Mapping......Page 16
3. Montel's Theorem......Page 20
4. The Hyperbolic Metric......Page 22
5. Quasiconformal Mappings......Page 26
6. Singular Integral Operators......Page 28
7. The Beltrami Equation......Page 30
1. Classification of Fixed Points......Page 38
2. Attracting Fixed Points......Page 42
3. Repelling Fixed Points......Page 43
4. Superattracting Fixed Points......Page 44
5. Rationally Neutral Fixed Points......Page 46
6. Irrationally Neutral Fixed Points......Page 52
7. Homeomorphisms of the Circle......Page 58
1. The Julia Set......Page 64
2. Counting Cycles......Page 69
3. Density of Repelling Periodic Points......Page 74
4. Polynomials......Page 76
1. Sullivan's Theorem......Page 80
2. The Classification Theorem......Page 85
3. The Wolif-Denjoy Theorem......Page 90
1. Siegel Disks......Page 92
2. Hyperbolicity......Page 100
3. Subhyperbolicity......Page 102
4. Locally Connected Julia Sets......Page 104
1. Polynomial-like Mappings......Page 110
2. Quasicircies......Page 112
3. Herman Rings......Page 114
4. Counting Herman Rings......Page 116
5. A Quasiconformal Surgical Procedure......Page 117
1. Invariant Spirals......Page 120
2. Repelling Arms......Page 124
3. John Domains......Page 128
1. The Mandeibrot Set......Page 134
2. The Hyperbolic Components of M......Page 144
3. Green's Function of Jc......Page 147
4. Green's Function of M......Page 150
5. External Rays with Rational Angles......Page 153
6. Misiurewicz Points......Page 159
7. Parabolic Points......Page 164
Epilogue......Page 172
References......Page 174
Index......Page 182
Symbol Index......Page 0
