Fractals and Chaos: The Mandelbrot Set and Beyond

321 Pages·2004·11.19 MB·English

by  Benoit B. Mandelbrot (auth.)

FRACTALS AND CHAOS
Springer Science+Business Media, LLC
SELECTED WORKS OF BENOIT B. MANDELBROT
REPRINTED, TRANSLATED, OR NEW
WITH ANNOTATIONS AND GUEST CONTRIBUTIONS
COMPANION TO THE FRACTAL GEOMETRY OF NATURE,
FRACTALS AND SCALING IN FINANCE (SELECTA E),
MULTIFRACTALS AND 1// NOISE (SELECTA N),
AND GAUSSIAN SELF-AFFINITY AND FRACTALS (SELECTA H)
Benoit B. Mandelbrot
FRACTALS AND
CHAOS
The Mandelbrot Set
and Beyond
SELECTA VOLUME C
with a foreword by P. W. Jones
and texts co-authored by
C.J.G. Bvertsz and M.C. Gutzwiller
, Springer
Benoit B. Mandelbrot
Mathematics Department
Yale University
New Haven, CT 06520-8283, USA
http://www.math.yale.edulmandelbrot
and
IBM T.J. Watson Research Center
Yorktown Heights, NY 10598-0218, USA
ISBN 978-1-4419-1897-0 ISBN 978-1-4757-4017-2 (eBook)
DOI 10.1007/978-1-4757-4017-2
Library of Congress Cataloging-in-Publication Data
Mandelbrot, Benoit, B.
Fractals and chaos : the Mandelbrot set and beyond / Benoit Mandelbrot.
p. cm.
Includes bibliographical references and index.
1. Fractals. 2. Mandelbrot sets. 2. Differentiable dynamical systems. 1. Title.
QA614.86.M232004
514'.742-dc22 2003063815
Printed on acid-free paper.
© 2004 Benoit B. Mandelbrot.
Originally published by Springer-Verlag New York, Inc. in 2004.
Softcover reprint of the hardcover 1s t edition 2004
All rights reserved. This work may not be translated or copied in whole or in part with
out the written permission of the publisher Springer Science+Business Media, LLC
except for brief excerots in connection with reviews or scholarly analysis. Use in connection
with any form of information storage and retrieval, electronic adaptation, computer software,
or by similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms,
even if they are not identified as such, is not to be taken as an expression of opinion as
to whether or not they are subject to proprietary rights.
9 8 7 6 5 4 3 2 1 SPIN 10962329
springeronline.com
My uncle Szolem Mandelbrojt (1899-1983)
initiated me-long ago-to the delights of iteration
To his memory I dedicate this intellectual fruit of mine
BENOIT B. MANDELBROT is Sterling He also received a Distinguished
Professor of Mathematical Sciences Service Award for Outstanding Achieve
at Yale University and IBM Fellow ment from the California Institute of
Emeritus (Physics) at the IBM Research Technology, and a Humboldt Preis from
Center. the Alexander von Humboldt Stiftung.
Best known as the founder of fractal Graduate of the Paris Ecole Polytech
geometry which is the first broad nique; M.S. and Ae.E. in Aeronautics,
attempt to investigate quantitatively the California Institute of Technology;
ubiquitous notion of roughness. Docteur es Sciences Mathematiques,
Author of Les objets fractals, 1975, Universite de Paris. Doctor honoris
1984, 1989, and 1995 (translated into causa: Syracuse University, Laurentian
Basque, Brazilian, Bulgarian, Chinese, University (Canada), Boston University,
Czech, Italian, Portugese, Rumanian, State University of New York, University
and Spanish) and The Fractal Geometry of of Guelph (Canada), University of Dallas,
Nature, 1982 (translated into Chinese, Union College, University of Buenos
German, Japanese, Korean, Russian, Aires (Argentina), Open University (UK),
and Spanish). His multi-volume Selecta Athens University of Business and
began with Fractals and Scaling in Commerce (Greece), University of St.
Finance: Discontinuity, Concentration, Risk, Andrews (Scotland), Emory University,
1997, Fractales, hasard et finance, 1997, Universitat Bremen (Germany), Pace
Multifractals and 11f Noise: Wild Self University, and the University of Tel
Affinity in PhYSiCS, 1999, and Gaussian Aviv (Israel).
Self-Affinity and Fractals, 2002. With M.L. Positions before joining IBM were
Frame, he authored Fractals, Graphics, with the CNRS in Paris, Philips
and Mathematics Education, 2002. Electronics, M.I.T., Princeton Institute for
Fellow, American Academy of Arts Advanced Study, University of Geneva,
and Sciences; Member U.S. National University of Lille, and Ecole Polyt ech
Academy of Sciences; Foreign Member, nique. Institute Lecturer at M.I.T.
Norwegian Academy of Science and Visiting Professor of Economics, later of
Letters. Applied Mathematics, Mathematics, and
His awards include the 1985 F. the Practice of Mathematics, at Harvard;
Barnard Medal for Meritorious Service to of Engineering at Yale; of Physiology at
Science ("Magna est Veritas"), granted by the Albert Einstein College of Medicine.
the U.S. National Academy of Sciences, Professeur de l'A cademie des Sciences
a
the 1986 Franklin Medal for Signal and l'Ecole Polyt echnique, Paris. Visited
Eminent Service in Science from the Cambridge, UK as c.c. Steward Visiting
Franklin Institute of Philadelphia, the Fellow at Conville and Caius College,
1988 Charles Proteus Steinmetz Medal from Scott Lecturer at Cavendish Laboratory
IEEE, the 1988 (first) Science for Art Prize and Member at Isaac Newton Institute of
from Moet-Hennessy-Louis Vuitton, Mathematical Sciences.
the 1989 Harvey Prize for Science and Mandelbrot had no formal teacher
Technology from the Technion in Haifa, but his early work was strongly influ
the 1991 Nevada Prize, the 1993 Wolf Prize enced by Paul Levy, Norbert Wiener,
for Physics, the 1994 Honda Prize, the 1996 and John von Neumann. An eloquent
Medaille de Vermeil de la Ville de Paris, the spokesman for "the unity of knowing
1999 John Scott Award, the 2000 Lewis and feeling," he seeks a measure of order
Fry Richardson Award of the European in physical, mathematical or social
Geophysical Society, the 2002 William phenomena that are characterized by
Procter Prize of Sigma Xi, and the 2003 abundant data but extreme sample
Japan Prize for Science and Technology. variability.
List of Chapters
In this List, * and (2003) mark the chapters not published previously. Other chapter titles
are followed (in parentheses) by the letter M followed by the year of publication and by the
lower case letter that the Bibliography uses to distinguish different texts published in the
same year. In the Bibliography, the items reproduced in this book and in Volumes E, N
and H are marked by a star followed by a chapter number, which in some cases is incom
plete.
*Foreword by Peter W. Jones (2003) ....................................................... ix
*Preface (2003) .............................................................................................. 1
I QUADRATIC JULIA AND MANDELBROT SETS
*CI Introduction to papers on quadratic dynamics:
a progression from seeing to discovering (2003) ................................... 9
*C2 Acknowledgments related to quadratic dynamics (2003) .................. 27
C3 Fractal aspects of the iteration of z ~ A z O-z)
for complex ,\ and z (M1980n) ................................................................ 37
C4 Cantor and Fatou dusts; self-squared dragons (M 1982F) ................ 52
C5 The complex quadratic map and its M-set (MI983p) ......................... 73
C6 Bifurcation points and the "n squared" approximation and
conjecture (MI985g), illustrated by M.L. Frame and K. Mitchell ..... 96
C7 The "normalized radical" of the M-set (MI985g) ............................... 100
C8 The boundary of the M-set is of dimension 2 (MI985g) .................. 110
C9 Certain Julia sets include smooth components (MI985g) ................. 114
CIO Domain-filling sequences of Julia sets,
and intuitive rationale for the Siegel discs (MI985g) ........................ 117
CII Continuous interpolation of the quadratic map and
intrinsic tiling of the interiors of Julia sets (MI985n) ........................ 125
viii LIST OF CHAYIERS ~ ~ CL
II NONQUADRATIC RATIONAL DYNAMICS
*C12 Introduction to chaos in nonquadratic dynamics: rational
functions devised from doubling formulas (2003) .. ........................... 137
C13 The map z -+ A. (z + 1/ z) and roughening of chaos from linear
to planar (computer-assisted homage to K. Hokusai) (MI984k) ..... 146
C14 Two nonquadratic rational maps, devised
from Weierstrass doubling formulas (1979-2003) ............................... 157
III ITERATED NONLINEAR FUNCTION SYSTEMS AND
THE FRACTAL LIMIT SETS OF KLEINIAN GROUPS
*C15 Introduction to papers on Kleinian groups,
their fractal limit sets, and IFS: history, recollections,
and acknowledgments (2003) ................................................................. 171
C16 Self-inverse fractals, Apollonian nets, and soap (M 1982F) ............. 178
C17 Symmetry by dilation or reduction, fractals, roughness (M2002w) 193
C18 Self-inverse fractals osculated by sigma-discs
and limit sets of inversion ("Kleinian") groups (MI983m) ............... 205
IV MULTIFRACTAL INVARIANT MEASURES
*C19 Introduction to measures that vanish exponentially
almost everywhere: DLA and Minkowski (2003) .............................. 221
C20 Invariant multifractal measures in chaotic Hamiltonian
systems and related structures (Gutzwiller & M 1988) .................... 231
C21 The Minkowski measure and multifractal anomalies in
invariant measures of parabolic dynamic systems (MI993s) ........... 239
C22 Harmonic measure on DLA and extended self-similarity
(M & Evertsz 1991) .................................................................................. 251
V BACKGROUND AND HISTORY
*C23 The inexhaustible function z squared plus c (1982-2003) ................. 259
*C24 The Fatou and Julia stories (2003) ......................................................... 268
*C25 Mathematical analysis while in the wilderness (2003) ...................... 276
CUMULATIVE BIBLIOGRAPHY .......................................................... 281
INDEX ........................................................................................................ 299
CF
Foreword by Peter W. Jones, Yale University
IT
IS ONLY TWENTY-THREE YEARS SINCE BENOIT MANDELBROT
published his famous picture of what is now called the Mandelbrot set.
The graphics available at that time seem primitive today, and
Mandelbrot's working drafts were even harder to interpret. But how that
picture has changed our views of the mathematical and physical universe!
Fractals, a term coined by Mandelbrot, are now so ubiquitous in the scien
tific consciousness that it is difficult to remember the psychological shock
of their arrival. A twenty-first-century researcher does not think twice
about using a computer simulation to begin the investigation of a
problem; indeed, it is now routine to use a desktop computer to search for
new phenomena or seek hints about research problems. In 1980 this was
very far from the case.
When a paradigm shift hits, it is rarely the old guard who ushers it in.
New methods are required, and accepted orthodoxy is often turned on its
head.
Thirty years ago, despite the appearance of an avant garde, there was
a general feeling in the mathematics community that one should distrust
pictures and any information they might carry. Computer experiments
had already appeared in the undergraduate physics curriculum, but were
almost nonexistent in mathematics. Perhaps this was due in part to the
relatively weak computers then available, but there were other aspects of
this attitude. Abstraction and generality were seen by many mathemati
cians as the guiding principles. There were cracks in this intellectual foun
dation, and the next twenty years were to see many of these prejudices
disappear.