Fractal Geometry and Analysis

485 Pages·1991·15.14 MB·English

by  Tim Bedford (auth.)

Fractal Geometry and Analysis
NATO ASI Series
Advanced Science Institutes Series
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Series C: Mathematical and Physical Sciences-Vol. 346
Fractal Geometry and Analysis
edited by
Jacques Belair
and
Serge Dubuc
Departement de mathematiques et de statistique,
Universite de Montreal,
Montreal, Quebec, Canada
Proceedings of the NATO Advanced Study Institute and
Seminaire de mathematiques superieures on
Fractal Geometry and Analysis
Montreal, Canada
July 3-21, 1989
Springer-Science+Business Media, B.V.
Proceedings of the NATO Advanced Study Institute and
Seminaire de mathematiques superieures on
Fractal Geometry and Analysis
Montreal, Canada
July 3-21, 1989
ISBN 978-94-015-7933-9 ISBN 978-94-015-7931-5 (eBook)
DOI 10.1007/978-94-015-7931-5
Printed on acid-free paper
Ali Rights Reserved
© 1991 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1991
Softcover re print of the hardcover 1s t edition 1991
No part of the material protected by this copyright notice may be reproduced or
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permission from the copyright owner.
Table of Contents
Preface vii
Participants ix
Contributors XV
Tim BEDFORD
Applications of dynamical systems theory to fractals - a study of cookie- 1
cutter Cantor sets
Paul BLANCHARD, Amy CHIU
Complex dynamics: an informal discussion 45
F. Michel DEKKING
Substitutions, branching processes and fractal sets 99
Serge DUBUC
Interpolation fractale 121
Kenneth J. FALCONER
Dimensions - their determination and properties 221
Masayoshi HATA
Topological a.Spects of self-similar sets and singular functions 255
Jean-Pierre KAHANE
Produits de poids aleatoires independants et applications 277
\
Michel MENDES FRANCE
The Planck constant of a curve 325
Claude TRICOT
Rectifiable and fractal sets 367
Edward R. VRSCAY
Iterated function systems: theory, applications and the inverse problem 405
Preface
This ASI-which was also the 28th session of the Seminaire de mathematiques superieures
of the Universite de Montreal - was devoted to Fractal Geometry and Analysis. The
present volume is the fruit of the work of this Advanced Study Institute. We were fortunate
to have with us Prof. Benoit Mandelbrot - the creator of numerous concepts in Fractal
Geometry - who gave a series of lectures on multifractals, iteration of analytic functions,
and various kinds of fractal stochastic processes.
Different foundational contributions for Fractal Geometry like measure theory, dy
namical systems, iteration theory, branching processes are recognized. The geometry of
fractal sets and the analytical tools used to investigate them provide a unifying theme of
this book. The main topics that are covered are then as follows.
Dimension Theory. Many definitions of fractional dimension have been proposed, all
of which coincide on "regular" objects, but often take different values for a given fractal
set. There is ample discussion on piecewise estimates yielding actual values for the most
common dimensions (Hausdorff, box-counting and packing dimensions). The dimension
theory is mainly discussed by Mendes-France, Bedford, Falconer, Tricot and Rata.
Construction of fractal sets. Scale in variance is a fundamental property of fractal sets.
It is often reflected in a functional equation satisfied by the coordinates of the points in the
sets. Both deterministic and random recurrent construction methods are presented, some
extending the classical notions of random walk and interpolation function. In another
instance, dynamical processes are involved in the study of generalized Cantor sets with
nonuniform scaling laws. All these models are described by Dekking, Bedford, Rata and
Vrscay.
Iterated Function Systems. The most novel scheme, which is also the most promising
one for applications in computer graphics and image reconstruction, namely "Iterated
Function Systems", is dealt with at length in a thoroughly transparent fashion by Vrscay.
The attractors of iterated function systems are also discussed by Falconer, Rata and
Dubuc.
Dynamics in the complex plane. Although geometrical in origin, fractal objects may
embody intrinsically dynamical aspects. Iteration of complex mappings is discussed by
Blanchard and major tools from the classical theory of one complex variable are shown
to be of substantial value.
Multifractals. The so-called "f( a )-spectrum" is presented in its natural setting, na
mely as a probability distribution rather than an ill-conceived "infinity of dimensions".
In this volume, it is primarily the paper by Kahane that deals with the subject of multi
fractals.
Applications of fractal geometry to mathematical analysis. There are many connections
vii
viii Preface
between fractal geometry, and mathematical analysis. Many challenging problems in the
field of analysis arise from fractal geometry, and there are many situations in analysis
where fractal geometry provides a new insight. This situation is illustrated by Kahane
(random coverings) and Dubuc (interpolation theory).
We feel that it is very important that mathematicians, especially those working in
mathematical analysis, contribute to fractal geometry. As the number and diversity of
scientists using fractal geometry increases (e.g., physicists, chemists, people in computer
science and computer graphics), more and more valuable applications can be expected
if there is a. significant input by mathematicians to the field of fractal geometry. Some
recent investigations in fractal geometry are presented in this book. We hope that these
will be helpful to other branches of science.
We wish to express our sincere thanks to all lecturers and participants for having
helped to make this ASIa success. Special thanks go to Aubert Daigneault, the Director
of the ASI, and Ghislaine David, its secretary, for their excellent organizational work.
The ASI was funded in large part by NATO, with additional support from the Natural
Sciences and Engineering Research Council of Canada., the Ministere de !'Education du
Quebec, and the Universite de Montreal. We would like to thank all these organizations
for their support. For their efforts on behalf of this ASI we are especially grateful to the
Scientific Affairs Division of NATO, particularly to Dr. Luis V. da. Cunha., the Director
of the ASI Programme.
Jacques Belair
Serge Dubuc
Scientific Directors
Participants
Simonetta ABENDA Paul BLANCHARD
Dipartimento di Fisica Department of Mathematics
Univ. degli Studi di Bologna Boston University
via Irnerio 46 Ill Cummington Street
40126 Bologna Boston, MA 02215
Italy U.S.A.
Alaaddin AL-DHAHIR Jose BRANDAO TIAGO
Faculty of Applied Mathematics Fac. de Ciencias Sociais e Humanas
University of Twente Universidade Nova de Lisboa
P.O. Box 217 A venida de Berna, 24
7500 AE Enschede 1000 Lisboa
The Netherlands Portugal
EminANARIM Carlos A. CABRELLI
Dept. of Electrical & Dept. of Electrical Engineering
Electronic Engineering & Computer Science
Bogazi~i University University of California
80815 Bebek-Istanbul Davis, CA 95616
Thrkey U.S.A.
Tim BEDFORD Robin C6TE
Mathematics and Informatics Departement de physique
Delft Univ. of Technology Universite Laval
Julianalaan 132 Cite Universitaire
2628 BL Delft Quebec, Quebec, GIK 7P4
The Netherlands Canada
Jacques BELAIR Anthony DAVIS
Departement de mathematiques Department of Physics
et de statistique McGill University
Universite de Montreal Ernest Rutherford Physics Bldg.
C.P. 6128, Succ. A 3600 University St.
Montreal, Quebec, H3C 3J7 Montreal, Quebec, H3A 2T8
Canada Canada
Fathi BEN NASR Michel DEKKING
Departement de mathematiques Mathematics and Informatics
Faculte des Sciences Delft Univ. of Technology
Universite de Thnisie Julianalann 132
5000 Monastir 2628 BL Delft
Thnisia The Netherlands
ix
X Participants
Gilles DESLAURIERS Norman FICKEL
Departement de matbematiques Mathematisches Institut
appliqmles Universitit Erlangen-Niirnberg
Ecole Polytechnique Bismarekstr. 1 1/2
Case Postale 6079, Succ. A W-8520 Erlangen
Montreal, Quebec, H3C 3A7 Germany
Canada
Jacqueline FLECKINGER
Serge DUBUC Mathematiques
Departement de mathtimatiques Universite Toulouse I
et de statistique Place Anatole France
Universite de Montreal 31042 Toulouse
C.P. 6128, Succ. A France
Montreal, Quebec, 113C 3J7
Canada Tony FOWLER
Department of Geology
Jean-Paul DUFOUR University of Ottawa
Institut de Mathematiques Ottawa, Ontario, K1N 6N5
Univ. des Sciences et Canada
Techniques de Languedoc
Pl. Eugene Bataillon Simon FRASER
34060 Montpellier Cedex Chemical Physics Theory Group
France Department of Chemistry
University of Toronto
Abdelkader ELQORTOBI Toronto, Ontario, M5S lAl
Departement de mathematiques Canada
Faculte des Sciences
Universite Mohamed 1er Erland GADDE
Oujda Department of Mathematics
Morocco University of Umea
S-901 87 Umea
Manuel L. ESQUIVEL Sweden
Departamento de Matematica
Univ. Nova de Lisboa Marcello GALEOTTI
Quinta da Torre 1st. di Matematica Applicata
2825 Monte da Caparica Universita di Firenze
Portugal Via Montebello, 7
50123 Firenze
Kenneth J. FALCONER Italy
School of Mathematics
University of Bristol Lucio GERONAZZO
Bristol BS8 1TW 1st. di Matematica Applicata
U.K. Universita di Firenze
Via Montebello, 7
Fan ai HUA 50123 Firenze
Mathematique, Batiment 425 Italy
Universite de Paris-Sud
Centre d'Orsay
91405 Orsay Gedex
France