Other Titles in This Series

51 Louis H. Kauffman, editor, The interface of knots and physics (San Francisco,

California, January 1995)

50 Robert Calderbank, editor, Different aspects of coding theory (San Francisco,

California, January 1995)

49 Robert L. Devaney , editor, Complex dynamical systems: The mathematics behind

the Mandlebrot and Julia sets (Cincinnati, Ohio, January 1994)

48 Walter Gautschi , editor, Mathematics of Computation 1943-1993: A half century of

computational mathematics (Vancouver, British Columbia, August 1993)

47 Ingrid Daubechies , editor, Different perspectives on wavelets (San Antonio, Texas,

January 1993)

46 Stefan A. Burr, editor, The unreasonable effectiveness of number theory (Orono,

Maine, August 1991)

45 D e W i t t L. Sumners , editor, New scientific applications of geometry and topology

(Baltimore, Maryland, January 1992)

44 Bela Bol lobas, editor, Probabilistic combinatorics and its applications (San

Francisco, California, January 1991)

43 Richard K. Guy , editor, Combinatorial games (Columbus, Ohio, August 1990)

42 C. Pomerance , editor, Cryptology and computational number theory (Boulder,

Colorado, August 1989)

41 R. W . Brockett , editor, Robotics (Louisville, Kentucky, January 1990) 40 Charles R. Johnson, editor, Matrix theory and applications (Phoenix, Arizona,

January 1989) 39 Robert L. Devaney and Linda Keen , editors, Chaos and fractals: The

mathematics behind the computer graphics (Providence, Rhode Island, August 1988) 38 Juris Hartmanis , editor, Computational complexity theory (Atlanta, Georgia,

January 1988) 37 Henry J. Landau, editor, Moments in mathematics (San Antonio, Texas, January

1987)

36 Carl de Boor , editor, Approximation theory (New Orleans, Louisiana, January 1986)

35 Harry H. Panjer, editor, Actuarial mathematics (Laramie, Wyoming, August 1985)

34 Michael Anshel and Wil l iam Gewirtz , editors, Mathematics of information

processing (Louisville, Kentucky, January 1984)

33 H. P e y t o n Young, editor, Fair allocation (Anaheim, California, January 1985) 32 R. W . McKelvey , editor, Environmental and natural resource mathematics (Eugene,

Oregon, August 1984) 31 B . Gopinath, editor, Computer communications (Denver, Colorado, January 1983) 30 S imon A. Levin, editor, Population biology (Albany, New York, August 1983)

29 R. A. DeMil lo , G. I. Davida, D . P. Dobkin, M. A. Harrison, and R. J. Lipton, Applied cryptology, cryptographic protocols, and computer security models (San Francisco, California, January 1981)

28 R. Gnanadesikan, editor, Statistical data analysis (Toronto, Ontario, August 1982)

27 L. A. Shepp, editor, Computed tomography (Cincinnati, Ohio, January 1982)

26 S. A. Burr, editor, The mathematics of networks (Pittsburgh, Pennsylvania, August 1981)

25 S. I. Gass, editor, Operations research: mathematics and models (Duluth, Minnesota, August 1979)

24 W . F. Lucas, editor, Game theory and its applications (Biloxi, Mississippi, January 1979)

(Continued in the back of this publication)

http://dx.doi.org/10.1090/psapm/049

AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures

published as a subseries of Proceedings of Symposia in Applied Mathematics

Proceedings of Symposia In

APPLIED MATHEMATICS

Volume 49

Complex Dynamical Systems The Mathematics Behind the Mandelbrot and Julia Sets

Robert L. Devaney, Editor

Bodil Branner Linda Keen Adrien Douady Paul Blanchard John H. Hubbard Dierk Schleicher Robert L. Devaney

[?, American Mathematical Society 15 Providence, Rhode Island

^?VDED

LECTURE NOTES PREPARED FOR THE

AMERICAN MATHEMATICAL SOCIETY SHORT COURSE

COMPLEX DYNAMICAL SYSTEMS

HELD IN CINCINNATI, OHIO

JANUARY 10-11, 1994

The AMS Short Course Series is sponsored by the Society's Program

Committee on National Meetings. The Series is under the direction of the

Short Course Subcommittee of the Program Committee

for National Meetings.

1991 Mathematics Subject Classification. Primary 58Fxx, 30Cxx.

Library of Congress Cataloging-in-Publicat ion D a t a

Complex dynamical systems: the mathematics behind the Mandelbrot and Julia sets / Robert L. Devaney, editor.

p. cm. — (Proceedings of symposia in applied mathematics, ISSN 0160-7634; v. 49) Includes bibliographical references and index. ISBN 0-8218-0290-9 (alk. paper). 1. Differentiable dynamical systems—Congresses. I. Devaney, Robert L., 1948- . II. Series.

QA614.8.C645 1994 514'.74—dc20 94-24190

CIP

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Assistant to the Publisher, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionQams.org.

© Copyright 1994 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Some articles were typeset using A\^S-T^i or A^S-WT^i, the American Mathematical Society's T]gX macro systems.

Pape r s in this volume are final versions, which will not be published elsewhere.

10 9 8 7 6 5 4 3 2 01 00 99 98 97 96

Contents

Preface ix

The Complex Dynamics of Quadratic Polynomials ROBERT L. DEVANEY 1

Puzzles and Para-Puzzles of Quadratic and Cubic Polynomials BODIL BRANNER 31

Julia Sets of Rational Maps LINDA KEEN 71

Does a Julia Set Depend Continuously on the Polynomial? ADRIEN DOUADY 91

The Dynamics of Newton's Method PAUL BLANCHARD 139

The Spider Algorithm JOHN H. HUBBARD AND DIERK SCHLEICHER 155

Complex Dynamics and Entire Functions ROBERT L. DEVANEY 181

Index 207

Preface

In the last fifteen years, the Mandelbrot set has emerged as one of the most recognizable objects in mathematics. While there is no question that this set is an object of great beauty, relatively few people appreciate the fact that the mathematics behind these images is equally beautiful. The goal of the Short Course at which these lectures were given was to remedy this.

The course was called Complex Dynamical Systems: The Mathematics Be­hind the Mandelbrot and Julia Sets and was held at the annual meeting of the American Mathematical Society in Cincinnati, Ohio, on January 10-11, 1994. The lectures covered a wide range of topics in complex dynamics. Among other topics, the lectures surveyed classical work of Julia and Fatou on local dynam­ics of anlaytic maps as well as more recent work on the dynamics of quadratic and cubic polynomials, the geometry of Julia sets, and the structure of vari­ous parameter spaces. The lectures contain reports on recent work on Yoccoz puzzles and tableaux, limiting dynamics near parabolic points, and the spider algorithm, as well as discussions of extensions of the theory to rational maps, Newton's method, and entire transcendental functions. Much of this book should be accessible to anyone with a background in the basics of dynamical systems and complex analysis.

Robert L. Devaney

Index

Ahlfors, 85, 88 Area zero, 87 Asymptotic value, 181 Attracting, 3, 76 Attracting axes, 97 Attracting basin, 78 Attracting cycles, 78, 96 Attracting petal, 80

Baker domains, 183 Baker, I. N., 2 Basin, 96, 98 Bers, 88 Bottcher coordinate, 37 Branner, 87 Brjuno, 82

Cantor bouquets, 186 Cantor set, 36 Caratheodory loop, 40 Caratheodory Theorem, 39 Cardioid, 42 Cayley, 144 Chaotic, 5 Chaotic behavior, 72 Co-critical point, 44 Collapsing trap, 104 Combinatorics, 86 Complex phase, 131 Conformally conjugate, 77 Conjugate, 8 Connectedness locus, 43 Covering maps, 73 Cremer, 83 Cremer points, 83 Critical annulus, 48, 55 Critical graph, 38, 45 Critical orbit, 3, 34 Critical orbit relation, 45 Critical piece, 47, 52 Critical points, 72, 96

free, 145 Critical position, 61, 64 Critical potential relation, 45

Critical tableau, 61 periodic, 62 recurrent, 62

Critical value, 73 Critical value piece, 47, 52 Critically intermittent recurrent, 64 Critically persistent recurrent, 64 Crown, 188 Cubic Fibonacci polynomial, 62 Curry, Garnett, and Sullivan, 144 Cycle, 96

Degree, 71, 97 Dendrite, 14 Devaney, 72, 87 Douady, 2, 81, 83, 86, 87 Douady and Hubbard, 144 Dynamic, 116 Dynamic [/] generated by / , 116

Eggbeater, 110 Eggbeater time, 115 Eigenvalue, 76 Entire transcendental function, 182 Equipotential, 37, 41 Escape locus, 44 Escape radius, 99 Escape rate, 36 Escape-time algorithm, 10 Eventually periodic, 84 Exceptional set, 75 Expanding, 72, 87 External argument, 39, 41, 202 External rays, 15, 17, 39, 41

Fatou, 83 Fatou component, 35 Fatou coordinates, 92, 116 Fatou set, 5, 34, 74 Fatou, Pierre, 2 Fibonacci dynamics, 65 Fibonacci grid, 62 Filled Julia set, 7, 34, 91, 99, 150, Finitely often renormalizable, 60

155

207

208

First return map, 76 Fixed points, 3 Fractal, 10 Full compact set, 36 Fundamental domain, 11

Generalized periodic point, 127 Green function, 36, 41 Grotzsch inequality, 46 Guckenheimer, 75

Hausdorff distance, 93 Hausdorff metric, 91 Herman, 83 Herman rings, 83 Holomorphic dynamical systems, 71 Horns map, 128 Hubbard, 2, 83, 86, 87 Hyperbolic components, 18, 40 Hyperbolic density, 86 Hyperbolic maps, 77, 85

co, 34 Immediate attracting basin, 78 Immediate basin, 96, 98 Indifferent cycles, 96 Infinitely often renormalizable, 60 Internal parametrization, 167 Invariant line field, 88 Irrational indifferent cycles, 96 Iterates, 71

Julia set, 5, 34, 72, 74, 91, 185 Julia, Gaston, 2

K-quasiconformal, 87 Kneading sequences, 157, 158

Lavaurs map, 125 Lifted phase, 123 Linearizable, 77, 98 Local behavior, 77 Local connectivity, 39 Locally connected, 60 Lower semi-continuous, 94 Lowest terms, 71

Mandelbrot, 2 Mandelbrot set, 13, 40, 86, 194

images, 148 limb of the, 43

M, 21, 156 Mane, 86 Mane-Sad-Sullivan Theorem, 107 Mafie-Sad-Sullivan decomposition, 45 McMullen, 87 Measurable Riemann mapping, 88 Method of divergence, 45 Misiurewicz point, 15

INDEX

Modulus, 46 Montel, 6, 74 Mori, 88 Multiplicity, 75 Multiplier, 96

Neutral, 3, 76 Neutral cycles, 96 Newton map, 139 Newton's method, 139

critical points for, 144 Non-critical position, 61 Normal, 6 Normal family, 73 Normal set, 74 Numerical algorithms, 139

Orbits, 3, 72

Para-puzzle, 56, 57 Para-puzzle-piece, 56, 57 Parabolic, 79 Parabolic chessboard, 120 Parabolic cycles, 96 Parabolic point, 110 Periodic, 73 Periodic cycle, 76 Periodic of period, 96 Polynomial-like mappings, 96, 148 Post-critrical, 73 Predictable, 74 Preperiodic, 3, 73 Proper, 95 Puzzle, 47 Puzzle-piece, 47, 52

Quadratic Fibonacci polynomial, 65 Quasiconformal maps, 87 Quasiconformally conjugate, 88

Rational map, 71 Rees, 86 Renomalized multipliers, 127 Renormalizable, 60 Renormalization, 128 Repelling, 3, 76 Repelling axes, 97 Repelling cycles, 96 Repelling periodic points, 185 Repelling petal, 80 Root point, 18, 40

Sad, 86 Schleicher's Algorithm, 23 Schoeder, 144 Schroder, 77 Semi-critical annulus, 55 Semi-critical position, 64

INDEX

Sensitive dependence on initial conditions, 4

Shishikura, 85 Siegel, 82 Siegel disks, 105 Siegel, C. L., 2 Singular values, 181 Spider algorithm, 155 Spider mapping, 164 Squaring function, 8 Stable, 74 Stable set, 5, 74 Standard annulus, 45 Standard spiders, 160 Structural instability, 189 Structural stability, 86 Structurally stable, 86 Sullivan, 85 Sullivan, Dennis, 2 Superattracting, 3, 76 Sutherland, 89

Tableau, 61 Tableau rules, 61, 64 Tableaux technique, 61 Thurston obstruction, 157 Thurston's Theorem, 177 Topologically conjuage, 86 Transitive, 5

Unimodal, 158 Unstable set, 74 Upper semi-continuous, 94

van Strien, S., 87 Virtual multipliers, 127

Wandering domains, 84, 183

Yoccoz, 31, 81, 82

Other Titles in This Series (Continued from the front of this publication)

23 R. V. Hogg, editor, Modern statistics: Methods and applications (San Antonio, Texas, January 1980)

22 G. H. Golub and J. Oliger, editors, Numerical analysis (Atlanta, Georgia, January

1978)

21 P. D . Lax, editor, Mathematical aspects of production and distribution of energy

(San Antonio, Texas, January 1976)

20 J. P. LaSalle, editor, The influence of computing on mathematical research and

education (University of Montana, August 1973)

19 J . T . Schwartz, editor, Mathematical aspects of computer science (New York City,

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18 H. Grad, editor, Magneto-fluid and plasma dynamics (New York City, April 1965)

17 R. Finn, editor, Applications of nonlinear partial differential equations in

mathematical physics (New York City, April 1964)

16 R. Bel lman, editor, Stochastic processes in mathematical physics and engineering

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15 N . C. Metropol is , A. H. Taub, J. Todd, and C. B . Tompkins, editors, Experimental arithmetic, high speed computing, and mathematics (Atlantic City and Chicago, April 1962)

14 R. Bel lman, editor, Mathematical problems in the biological sciences (New York

City, April 1961)

13 R. Bel lman, G. Birkhoff, and C. C. Lin, editors, Hydrodynamic instability (New

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12 R. Jakobson, editor, Structure of language and its mathematical aspects (New York

City, April 1960)

11 G. Birkhoff and E. P. Wigner , editors, Nuclear reactor theory (New York City,

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10 R. Bel lman and M. Hall, Jr., editors, Combinatorial analysis (New York

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9 G. Birkhoff and R. E. Langer, editors, Orbit theory (Columbia University, April

1958)

8 L. M. Graves, editor, Calculus of variations and its applications (University of

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7 L. A. MacColl , editor, Applied probability (Polytechnic Institute of Brooklyn, April

1955)

6 J. H. Curtiss, editor, Numerical analysis (Santa Monica City College, August 1953)

5 A. E. Heins, editor, Wave motion and vibration theory (Carnegie Institute of

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4 M. H. Martin, editor, Fluid dynamics (University of Maryland, June 1951)

3 R. V. Churchill, editor, Elasticity (University of Michigan, June 1949)

2 A. H. Taub, editor, Electromagnetic theory (Massachusetts Institute of Technology, July 1948)

1 E. Reissner, editor, Non-linear problems in mechanics of continua (Brown University, August 1947)

(See the AMS catalog for earlier titles)