Complex Dynamics

181 Pages·1993·14.089 MB·English

by  Lennart Carleson, Theodore W. Gamelin (auth.)

Universitext: Tracts in Mathematics
Editorial Board
(North America):
S. Axler
F.W. Gehring
P.R. Halmos
Universitext
Editors (North America): S. Axler, F.W. Gehring, and P.R. Halmos
Aksoy/Khamsi: Nonstandard Methods in Fixed Point Theory
Aupetit: A Primer on Spectral Theory
BoosslBleecker: Topology and Analysis
Borkar: Probability Theory; An Advanced Course
CarlesonlGamelin: Complex Dynamics
Cecil: Lie Sphere Geometry: With Applications to Submanifolds
Chae: Lebesgue Integration (2nd ed.)
Charlap: Bieberbach Groups and Flat Manifolds
Chern: Complex Manifolds Without Potential Theory
Cohn: A Classical Invitation to Algebraic Numbers and Cla~s Fields
Curtis: Abstract Linear Algebra
Curtis: Matrix Groups
DiBenedetto: Degenerate Parabolic Equations
Dimca: Singularities and Topology of Hypersurfaces
Edwards: A Formal Background to Mathematics I alb
Edwards: A Formal Background to Mathematics II alb
Foulds: Graph Theory Applications
Gardiner: A First Course in Group Theory
Garding!Tambour: Algebra for Computer Science
Goldblatt: Orthogonality and Spacetime Geometry
Hahn: Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups
Holmgren: A First Course in Discrete Dynamical Systems
HoweITan: Non-Abelian Harmonic Analysis: Applications of SL(2, R)
Howes: Modem Analysis and Topology
HumiIMiller: Second Course in Ordinary Differential Equations
HurwitzIKritikos: Lectures on Number Theory
Jennings: Modern Geometry with Applications
Jones/MorrislPearson: Abstract Algebra and Famous Impossibilities
KannanlKrueger: Advanced Real Analysis
KellylMatthews: The Non-Euclidean Hyperbolic Plane
Kostrikin: Introduction to Algebra
Luecking/Rubel: Complex Analysis: A Functional Analysis Approach
MacLanelMoerdijk: Sheaves in Geometry and Logic
Marcus: Number Fields
McCarthy: Introduction to Arithmetical Functions
Meyer: Essential Mathematics for Applied Fields
Mines/RichmanIRuitenburg: A Course in Constructive Algebra
Moise: Introductory Problems Course in Analysis and Topology
Morris: Introduction to Game Theory
Porter/Woods: Extensions and Absolutes of Hausdorff Spaces
RamsaylRichtmyer: Introduction to Hyperbolic Geometry
Reisel: Elementary Theory of Metric Spaces
Rickart: Natural Function Algebras
Rotman: Galois Theory
Sagan: Space-Filling Curves
(continued after index)
Lennart Carleson Theodore W. Gamelin
Complex Dynamics
With 28 Figures
, Springer
Lennart Carleson Theodore W. Gamelin
Department of Mathematics Department of Mathematics
Royal Institute of Technology University of California
S-loo 44 Stockholm Los Angeles, CA 90024-1555
Sweden USA
and Department of Mathematics
University of California
Los Angeles, CA 90024-1555
USA
Editorial Board (North America):
S. Axler F. W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Michigan State University Universtiy of Michigan Santa Clara University
East Lansing, MI 48824 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
On the cover: A filled-in Julia set with parabolic fixed point, attracting petals, and repelling
arms.
Mathematics Subject Classification (1991): 3OCxx, S8Fxx
Library of Congress Cataloging-in-Publication Data
Carleson, Lennart.
Complex dynamics/by L. Carleson and T. Gamelin.
p. cm. - (Universitext)
Includes bibliographical references and index.
ISBN-13: 978-0-387-97942-7 e-ISBN-13: 978-1-4612-4364-9
001: 10.1007/978-1-4612-4364-9
I. Functions of complex variables. 2. Mappings (Mathematics)
3. Fixed point theory. I. Title.
QA33 1.7 . C37 1993
SIS'.9-dc20 92-32457
Printed on acid-free paper.
© 1993 Springer-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the
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Production managed by Francine McNeill; manufacturing supervised by Vincent Scelta.
Photocomposed copy prepared from the author's .A,A.1S-1EX file.
9 8 7 6 5 4 3 2 (Corrected second printing, 1995)
Preface
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. We would
vi Preface
particularly like to acknowledge contributions from Peter Jones and
M. Shishikura. Any reader familiar with the area will recognize the
exposition of quasiconformal mappings from Ahlfors' book. It is often
difficult to trace particular results to the rightful owners, particularly
in such a rapidly developing area where so much seems to flow by
word of mouth. We apologize for any inadequacy and for omissions.
L. Carleson and T.W. Gamelin
Los Angeles, March, 1992
Contents
Preface v
I. Conformal and Quasiconformal Mappings 1
1. Some Estimates on Conformal Mappings . 1
2. The Riemann Mapping. 5
3. Montel's Theorem .... 9
4. The Hyperbolic Metric . . 11
5. Quasiconformal Mappings 15
6. Singular Integral Operators 17
7. The Beltrami Equation. . . 19
II. Fixed Points and Conjugations 27
1. Classification of Fixed Points 27
2. Attracting Fixed Points ... 31
3. Repelling Fixed Points .... 32
4. Superattracting Fixed Points 33
5. Rationally Neutral Fixed Points. 35
6. lrrat ionally Neutral Fixed Points 41
7. Homeomorphisms of the Circle 47
viii Contents
III. Basic Rational Iteration 53
1. The Julia Set . . . . . . . 53
2. Counting Cycles .......... . 58
3. Density of Repelling Periodic Points 63
4. Polynomials . . . . . . . . . . . . . . 65
IV. Classification of Periodic Components 69
1. Sullivan's Theorem ..... 69
2. The Classification Theorem 74
3. The Wolff-Denjoy Theorem 79
V. Critical Points and Expanding Maps 81
1. Siegel Disks . . . 81
2. Hyperbolicity . . . . . . . . . 89
3. Subhyperbolicity . . . . . . . 91
4. Locally Connected Julia Sets 93
VI. Applications of Quasiconformal Mappings 99
1. Polynomial-like Mappings 99
2. Quasicircles ...... . .101
3. Herman Rings. . . . . . . .103
4. Counting Herman Rings. .105
5. A Quasiconformal Surgical Procedure .106
VII. Local Geometry of the Fatou Set 109
1. Invariant Spirals .109
2. Repelling Arms .113
3. John Domains. . .117
VIII. Quadratic Polynomials 123
1. The Mandelbrot Set . . . 123
2. The Hyperbolic Components of M .133
3. Green's Function of Jc . . . . . . . .136
4. Green's Function of M . . . . . . . .139
5. External Rays with Rational Angles .142
6. Misiurewicz Points .148
7. Parabolic Points .......... . . 153
Contents ix
Epilogue 161
References 163
Index 171
Symbol Index 175