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Complex Dynamics : Universitext / Universitext: Tracts in Mathematics - Lennart Carleson

Complex Dynamics

Universitext / Universitext: Tracts in Mathematics

Paperback Published: 2nd February 1996
ISBN: 9780387979427
Number Of Pages: 192

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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.

Conformal and Quasiconformal Mappingsp. 1
Some Estimates on Conformal Mappingsp. 1
The Riemann Mappingp. 5
Montel's Theoremp. 9
The Hyperbolic Metricp. 11
Quasiconformal Mappingsp. 15
Singular Integral Operatorsp. 17
The Beltrami Equationp. 19
Fixed Points and Conjugationsp. 27
Classification of Fixed Pointsp. 27
Attracting Fixed Pointsp. 31
Repelling Fixed Pointsp. 32
Superattracting Fixed Pointsp. 33
Rationally Neutral Fixed Pointsp. 35
Irrationally Neutral Fixed Pointsp. 41
Homeomorphisms of the Circlep. 47
Basic Rational Iterationp. 53
The Julia Setp. 53
Counting Cyclesp. 58
Density of Repelling Periodic Pointsp. 63
Polynomialsp. 65
Classification of Periodic Componentsp. 69
Sullivan's Theoremp. 69
The Classification Theoremp. 74
The Wolff-Denjoy Theoremp. 79
Critical Points and Expanding Mapsp. 81
Siegel Disksp. 81
Hyperbolicityp. 89
Subhyperbolicityp. 91
Locally Connected Julia Setsp. 93
Applications of Quasiconformal Mappingsp. 99
Polynomial-like Mappingsp. 99
Quasicirclesp. 101
Herman Ringsp. 103
Counting Herman Ringsp. 105
A Quasiconformal Surgical Procedurep. 106
Local Geometry of the Fatou Setp. 109
Invariant Spiralsp. 109
Repelling Armsp. 113
John Domainsp. 117
Quadratic Polynomialsp. 123
The Mandelbrot Setp. 123
The Hyperbolic Components of Mp. 133
Green's Function of Tcp. 136
Green's Function of Mp. 139
External Rays with Rational Anglesp. 142
Misiurewicz Pointsp. 148
Parabolic Pointsp. 153
Epiloguep. 161
Referencesp. 163
Indexp. 171
Symbol Indexp. 175
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387979427
ISBN-10: 0387979425
Series: Universitext / Universitext: Tracts in Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 192
Published: 2nd February 1996
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.47 x 15.62  x 1.32
Weight (kg): 0.33