FRACTAL GEOMETRY Mathematical Foundations and Applications Fractal Geometry: Mathematical Foundations and Application Second Edition Kenneth Falconer  2003 John Wiley & Sons, Ltd ISBNs: 0-470-84861-8 (HB); 0-470-84862-6 (PB) FRACTAL GEOMETRY Mathematical Foundations and Applications Second Edition Kenneth Falconer University of St Andrews, UK Copyright  2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620 This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-470-84861-8 (Cloth) ISBN 0-470-84862-6 (Paper) Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production Contents Preface ix Preface to the second edition xiii Course suggestions xv Introduction xvii Notes and references xxvii PART I FOUNDATIONS Chapter Mathematical background 1.1 1.2 1.3 1.4 1.5 Chapter Basic set theory Functions and limits Measures and mass distributions Notes on probability theory Notes and references Exercises Hausdorff measure and dimension 2.1 Hausdorff measure 2.2 Hausdorff dimension 2.3 Calculation of Hausdorff dimension—simple examples *2.4 Equivalent definitions of Hausdorff dimension *2.5 Finer definitions of dimension 2.6 Notes and references Exercises Chapter 27 Alternative definitions of dimension 3.1 3.2 11 17 24 25 Box-counting dimensions Properties and problems of box-counting dimension 27 31 34 35 36 37 37 39 41 47 v vi Contents *3.3 Modified box-counting dimensions *3.4 Packing measures and dimensions 3.5 Some other definitions of dimension 3.6 Notes and references Exercises Chapter Densities Structure of 1-sets Tangents to s-sets Notes and references Exercises Chapter Projections of arbitrary sets Projections of s-sets of integral dimension Projections of arbitrary sets of integral dimension Notes and references Exercises 59 68 70 73 74 74 76 76 80 84 89 89 90 90 93 95 97 97 99 Intersections of fractals 49 50 53 57 57 59 Product formulae Notes and references Exercises 8.1 Intersection formulae for fractals *8.2 Sets with large intersection 8.3 Notes and references Exercises PART II Products of fractals 7.1 7.2 Chapter Projections of fractals 6.1 6.2 6.3 6.4 Chapter Local structure of fractals 5.1 5.2 5.3 5.4 Chapter Techniques for calculating dimensions 4.1 Basic methods 4.2 Subsets of finite measure 4.3 Potential theoretic methods *4.4 Fourier transform methods 4.5 Notes and references Exercises Chapter 99 107 107 109 APPLICATIONS AND EXAMPLES Iterated function systems—self-similar and self-affine sets 9.1 Iterated function systems 9.2 Dimensions of self-similar sets 110 113 118 119 121 123 123 128 vii 9.3 Some variations 9.4 Self-affine sets 9.5 Applications to encoding images 9.6 Notes and references Exercises Chapter 10 Examples from number theory 10.1 10.2 10.3 10.4 Distribution of digits of numbers Continued fractions Diophantine approximation Notes and references Exercises 151 Chapter 11 Graphs of functions 11.1 Dimensions of graphs *11.2 Autocorrelation of fractal functions 11.3 Notes and references Exercises General theory of Julia sets Quadratic functions—the Mandelbrot set Julia sets of quadratic functions Characterization of quasi-circles by dimension Newton’s method for solving polynomial equations Notes and references Exercises 176 179 181 182 184 185 186 Chapter 14 Iteration of complex functions—Julia sets 14.1 14.2 14.3 14.4 14.5 14.6 160 169 173 173 176 Chapter 13 Dynamical systems 13.1 Repellers and iterated function systems 13.2 The logistic map 13.3 Stretching and folding transformations 13.4 The solenoid 13.5 Continuous dynamical systems *13.6 Small divisor theory *13.7 Liapounov exponents and entropies 13.8 Notes and references Exercises 151 153 154 158 158 160 Chapter 12 Examples from pure mathematics 12.1 Duality and the Kakeya problem 12.2 Vitushkin’s conjecture 12.3 Convex functions 12.4 Groups and rings of fractional dimension 12.5 Notes and references Exercises 135 139 145 148 149 187 189 193 198 201 205 208 211 212 215 215 223 227 235 237 241 242 viii Contents Chapter 15 Random fractals 15.1 15.2 15.3 A random Cantor set Fractal percolation Notes and references Exercises 244 Chapter 16 Brownian motion and Brownian surfaces 16.1 16.2 16.3 16.4 16.5 Brownian motion Fractional Brownian motion ´ stable processes Levy Fractional Brownian surfaces Notes and references Exercises 258 Chapter 17 Multifractal measures 17.1 17.2 17.3 17.4 Coarse multifractal analysis Fine multifractal analysis Self-similar multifractals Notes and references Exercises Fractal growth Singularities of electrostatic and gravitational potentials Fluid dynamics and turbulence Fractal antennas Fractals in finance Notes and references Exercises 258 267 271 273 275 276 277 Chapter 18 Physical applications 18.1 18.2 18.3 18.4 18.5 18.6 246 251 255 256 278 283 286 296 296 298 300 306 307 309 311 315 316 References 317 Index 329 Preface I am frequently asked questions such as ‘What are fractals?’, ‘What is fractal dimension?’, ‘How can one find the dimension of a fractal and what does it tell us anyway?’ or ‘How can mathematics be applied to fractals?’ This book endeavours to answer some of these questions The main aim of the book is to provide a treatment of the mathematics associated with fractals and dimensions at a level which is reasonably accessible to those who encounter fractals in mathematics or science Although basically a mathematics book, it attempts to provide an intuitive as well as a mathematical insight into the subject The book falls naturally into two parts Part I is concerned with the general theory of fractals and their geometry Firstly, various notions of dimension and methods for their calculation are introduced Then geometrical properties of fractals are investigated in much the same way as one might study the geometry of classical figures such as circles or ellipses: locally a circle may be approximated by a line segment, the projection or ‘shadow’ of a circle is generally an ellipse, a circle typically intersects a straight line segment in two points (if at all), and so on There are fractal analogues of such properties, usually with dimension playing a key rˆole Thus we consider, for example, the local form of fractals, and projections and intersections of fractals Part II of the book contains examples of fractals, to which the theory of the first part may be applied, drawn from a wide variety of areas of mathematics and physics Topics include self-similar and self-affine sets, graphs of functions, examples from number theory and pure mathematics, dynamical systems, Julia sets, random fractals and some physical applications There are many diagrams in the text and frequent illustrative examples Computer drawings of a variety of fractals are included, and it is hoped that enough information is provided to enable readers with a knowledge of programming to produce further drawings for themselves It is hoped that the book will be a useful reference for researchers, providing an accessible development of the mathematics underlying fractals and showing how it may be applied in particular cases The book covers a wide variety of mathematical ideas that may be related to fractals, and, particularly in Part II, ix References 323 Kahane J.-P (1985) Some Random Series of Functions, 2nd edition, Cambridge University Press, Cambridge Kahane J.-P (1986) Sur la dimensions des intersections, in Aspects of Mathematics and its Applications (Ed J A Barroso), pp 419–430, North-Holland, Amsterdam Kahane J.-P and Peyri`ere J (1976) Sur certaines martingales de Benoit Mandelbrot, Adv Math 22, 131–145 Karlin S and Taylor H M (1975) A First Course in Stochastic Processes, Academic Press, New York Karlin S and Taylor H M (1981) A Second Course in Stochastic Processes, Academic Press, New York Katok A and Hasselblatt B (1995) Introduction to the Modern Theory of 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dynamical system 187, 189, 192, 194 autocorrelation 169–173, 268 autocorrelation function 170 average 21 Avogadro’s number 258 baker’s transformation 193, 193–194, 194, 210, 211 ball Banach’s contraction mapping theorem 124, 125 basic interval 35, 62, 127, 152 basin of attraction 187, 201, 222, 238–241 Besicovitch construction 178, 179 Besicovitch set 179 bifurcation diagram 192 bijection 7, 236 bi-Lipschitz equivalence 236 bi-Lipschitz function 8, 103 bi-Lipschitz transformation 32, 33, 41 binary interval 16, 36 Borel set(s) and duality 178 and intersections 111 measure on 11–12 and products 100, 101, 105, 106 and projections 90, 93 and subrings 183–184 and subsets of finite measure 68, 69, 70 boundary 5, bounded set 5, 114 bounded variation 86 box counting 283, 298 box(-counting) dimension 41, 41–47, 43, 44 Brownian motion 263, 265 fractional Brownian motion 268 fractional Brownian surface 273 and growth 299–300 and Hausdorff dimension 46, 60 L´evy stable process 272 limitations 48 Fractal Geometry: Mathematical Foundations and Application Second Edition Kenneth Falconer  2003 John Wiley & Sons, Ltd ISBNs: 0-470-84861-8 (HB); 0-470-84862-6 (PB) 329 330 box(-counting) dimension (continued ) modified 49, 49–50 problems 48–49 properties 47–48 ways of finding 44, 47 branch 223 branching process theory 250 Brownian graph 266 Brownian motion 37, 258–267, 259, 261, 298, 299, 311–315 Fourier series representation 260 fractional 173, 267, 267–271, 269, 275 index-α fractional 267, 267–271, 269, 311 multifractional 271 n-dimensional 260 Brownian sample function 259, 261, 262 graph 265–266 index-α fractional 273 multiple points 265 Brownian surface, fractional 273–275 Brownian trail(s) 258, 261, 262, 264–265 Cantor dust xx, xxi, 94, 126 calculation of Hausdorff dimension 34 construction of xxi , 34 ‘Cantor product’ 99, 100, 102, 103 Cantor set as attractor 189, 192, 194 construction of 61–62 dimension of xxiii, 34–35, 47 middle λ 64 middle third xvii, xviii, xviii , 34–35, 47, 60, 61, 63–64, 87, 99, 100, 112, 123–124, 127, 129, 188, 189 non-linear 136–137, 154 random 245–246, 246–250, 256 uniform 63–64, 64, 103, 141 see also main entry: middle third Cantor set ‘Cantor target’ 103–104, 104 (s-)capacity 72 capacity dimension 41 Cartesian product 4, 87, 99, 100 cat map 213 Cauchy sequence 125 central limit theorem 24, 259 chaos 189 chaotic attractor 189, 192 chaotic repeller 189 characteristic function 271 Choleski decomposition 270 Index circle perimeter, dimension print 56 class Cs 113, 113–118, 157 member 113 closed ball 4, 66 closed interval closed loop, as attractor 201 closed set 5, 5, 114, 201 closure of set cloud boundaries 244, 298 coarse multifractal spectrum 278–279 lower 280 upper 280 coastline(s) 54, 244, 298 codomain collage theorem 145 compact set 6, 48 complement of a set complete metric 125 complex dynamics 216 composition of functions computer drawings 127–128, 128, 145–148, 196, 227, 232, 233, 234, 239, 240, 244, 260, 269, 270–271, 275 conditional expectation 22 conditional probability 19 conformal mapping 138–139, 180, 241 congruence(s) 7, 7, 112 direct conjugacy 223 connected component connected set content Minkowski 45 s-dimensional 45 continued fraction(s) 153–154 examples 153 continued fraction expansion 153 continuity equation 203 continuous dynamical systems 201–205 continuous function 10 continuously differentiable function 10 contours 275 contracting similarity 123 contraction 123 contraction mapping theorem 124, 125 convergence pointwise 10 of sequence uniform 10, 17 convex function 181, 181–182, 287 convex surface 182 convolution theorem 73, 172 coordinate cube Index copper sulphate, electrolysis of 300–301, 302–304 correlation 169–173, 268, 311 countable set 4, 32, 41 countable stability of dimension print 55 of Hausdorff dimension 32, 41 counting measure 13 covariance matrix 270 cover of a set 27, 28, 30, 35–36 see also (δ-)cover covering lemma 66–67 critical exponent 54 critical point 227, 228 cross section 202 cube 4, 42 cubic polynomial, Newton’s method for 239–241, 240 curve 53, 81 fractal 133–134 Jordan 53, 81 rectifiable 81, 81–84, 85–86, 181–182 curve-free set 81, 83, 83–84 curve-like set 81, 82–83, 83–84 Darcy’s law 305 data compression 145–148 decomposition of 1-sets 80–81, 81 (delta-)/(δ-)cover 27, 28, 30, 36 (delta-)/(δ-)neighbourhood 4, 4, 45, 112, 124 (delta-)/(δ-)parallel body dendrite 232 dense set density 76, 76–80, 84, 88, 152 lower 77, 84 upper 77 density function 306 derivative 10 diameter of subset 5, 27 difference of sets differentiability 10, 137–138, 160, 182 continuous 10 diffusion equation 303 diffusion limited aggregation (DLA) model 301–306, 302, 303 digital sundial 96–97, 97 dimension xxii–xxv, xxiv alternative definitions 39–58 approximations to 299, 305 of attractors and repellers 186–201 box(-counting) 41, 41–50, 43, 44, 60, 263, 265, 268, 272, 273, 299–300 331 calculation of 34–35, 59–75, 128–135 of Cantor set xxiii capacity 41 characterization of quasi-circles by 235–237 divider 39, 53–54 entropy 41 experimental estimation of 39–40, 299–300 finer definitions 36–37 Fourier 74 of graphs of functions 160–169, 161 Hausdorff xxiv, 27, 31, 31–33, 35–36, 40, 54, 70, 92 Hausdorff–Besicovitch 31 Hausdorff dimension of a measure 209, 288 information 41 of intersections 109–118 local 283 lower box(-counting) 41, 43 of a measure 288 metric 41 Minkowski(–Bouligand) 46 modified box-counting 49, 49–50 one-sided 54 packing 50–53, 51 of products 99–107 of projections 90–97 of random sets 246–251, 259–275 of self-affine sets 106–107, 139–144, 166–169 of self-similar sets xxiv, 128–135 similarity xxiv topological xxv upper box(-counting) 41, 43 of von Koch curve xxiii of (α-)well approximable number 155 dimension function 37 dimension print 54–57, 55, 56 disadvantages 55–56 examples 56 dimensionally homogeneous set 50 Diophantine approximation 153, 154–157, 205 direct congruence Dirichlet’s theorem 154 discrete dynamical system 186, 186–201 disjoint binary intervals 36 disjoint collection of sets distance set 183 332 distribution Gaussian 23 multidimensional normal 262 normal 23, 260 uniform 23 distribution of digits 151–153 divider dimension 39, 53–54 domain double sector 85 duality method 176–179, 177 ‘dust-like’ set 110 dynamical systems 186–212 continuous 201–205 discrete 186, 186–201 dynamics, complex 216 eddies 308 Egoroff’s theorem 17, 79 electrical discharge in gas 305 electrodynamics 310 electrolysis 300–301, 301, 302–304 electrostatic potential 70, 305, 306–307 (s-)energy 70 entropy 209–211, 210 entropy dimension 41 ergodic theory 128, 209 Euclidean distance or metric Euclidean space event 18 independence of 20 event space 18 exchange rate variation 311, 313 expectation 21 conditional 22 expectation equation 247, 250 experiment (probabilistic) 18 experimental approach to fractals 39–40, 299–300 exterior, of loop 223 extinction probability 250 Fatou set 216, 235 Feigenbaum constant 193 figure of eight 223, 228 filled-in Julia set 215 finance 186, 311–315 fine (Hausdorff) multifractal spectrum 284 fine multifractal analysis 277, 283–286 first return map 202, 202 fixed point 125, 186, 216 fluid dynamics 277, 307–309 forked lightning 298, 300 Index Fourier dimension 74 Fourier series 206 Fourier transform 73, 92 methods using 73–74, 112, 171–173, 260, 271 fractal, definition xxii, xxv fractal curve 133–134 , 216 fractal growth 300–306 fractal interpolation 169, 170 fractal percolation 251–255 fractally homogeneous turbulence 309 fractional Brownian motion 173, 267, 269, 267–271 fractional Brownian surface(s) 273–275 fractions, continued 153–154 Frostman’s lemma 70 full square 254 function(s) 6, 6–7 continuous 10 convex 181, 181–182 functional analysis 179 gauge function 37 Gaussian distribution 23 Gaussian process 267 general construction 61–62, 62 generator 134–135 examples 132, 133, 134 geometric invariance 41 geometric measure theory 53, 76 graphs of functions 160, 160–169, 258, 266, 267 gravitational potential 70, 306–307 group(s) of fractional dimension 182–184 group of transformations 8, 110, 111 growth 300–306 Hamiltonian 207–208 Hamiltonian systems, stability of 207–208, 212 Hamilton’s equations 207 Hausdorff–Besicovitch dimension 31 Hausdorff dimension xxiv, 27, 31, 31–33, 31, 54 of attractor 192, 193 and box(-counting) dimension 46, 60 Brownian motion 263, 265 calculating 70–72, 92 equivalent definitions 35–36 fractional Brownian motion 268 fractional Brownian surface 273 L´evy stable process 272 Index of a measure 209, 288 and packing dimension 53 and projections 90–93 properties 32 of self-affine set 140–144, 140, 144 Hausdorff distance 124, 124 Hausdorff measure 14, 27–30, 28 and intersections 113 and packing measure 53 and product rule 99 and quasi-circle 236 Hausdorff metric 124, 124, 145, 300 heat equation 303 Hele–Shaw cell 304 H´enon attractor/map 103, 196–197, 197, 198, 212, 213 heuristic calculation(s) 3435, 129 histogram method, for multifractal spectrum 279, 283 Hăolder condition 30, 32, 262, 265, 268 Hăolder exponent 283, 312 Hăolder function 8, 10, 30, 161 Hăolders inequality 297 homeomorphism 10, 33 homogeneous turbulence 308 horseshoe map 194–195, 195, 196, 212 image 7, 258, 275 image encoding 145–148 in general 109, 110 independence of events 20 of random variables 20 independent increment 259, 268, 311 index-α fractional Brownian function 273 index-α fractional Brownian motion 267, 267–271, 269, 311–312 index-α fractional Brownian surface 273, 274 infimum information dimension 41 injection integral 16–17 integral geometry 118 interior of loop 223 intermittency 308 interpolation 169, 170 intersection formulae 110–113 intersection(s) 4, 109–118, 110, 265–266, 275 large 113–118, 157 interval 333 interval density 152 invariance geometric 41 Lipschitz 41, 48 invariant curves 205, 207 invariant measure 208, 208–211 invariant set 123, 123–129, 187, 218 invariant tori 208 inverse function inverse image investment calculations 186 irregular point 78 irregular set 78, 79–80, 94 examples xxi , 81, 180 isolated point, as attractor 201 isometry 7, 30 isotropic 261 isotropic turbulence 307 iterated construction(s) 95–96, 96, 180–181 iterated function system (IFS) 123, 123–128 advantages 128 attractor for 123, 123–129, 146–148, 194, 228 and repellers 187–189 variations 135–139 iterated Venetian blind construction 95–96, 96, 180 iteration 186–201, 215–242 Jarn´ık’s theorem 155–157, 205, 207 Jordan curve 53, 81 Julia set xxii, 215, 215–242, 219, 233, 234 Kakeya problem 176–179 Koch curve see von Koch curve Kolmogorov–Arnold–Moser (KAM) theorem 208 Kolmogorov entropy 41 Kolmogorov model of turbulence 307–309 laminar flow 307 landscapes 273 Laplace’s equation 304, 305 law of averages 23–24 Lebesgue density theorem 77, 77, 93 Lebesgue measure 13, 16, 112, 192, 264 n-dimensional 13, 17, 28, 112, 143, 266 Legendre spectrum 282 334 Legendre transform 281, 281, 282, 287 length 13, 81 scaling of 29 level set 266, 275 level-k interval 35, 62, 152 level-k set 127 L´evy process 267, 271–273 L´evy stable process 271, 271–273 Liapounov exponents 208–211, 209, 210, 212 limit 8–9 lower of sequence upper lim sup sets 113 line segment dimension print 56 uniform mass distribution on 14 line set 176, 176–179 linear transformation Lipschitz equivalence 236 Lipschitz function 8, 10, 30, 103 Lipschitz invariance 41, 48, 55, 56 Lipschitz transformation 30, 32, 34 local dimension 283 local product 103, 196, 202 local structure of fractals 76–89 logarithmic density 41 logarithms 10 logistic map 189–193, 191, 192, 212 long range dependence 311 loop 223, 224–225 closed 201 Lorenz attractor 203–204, 204 Lorenz equations 202–203 lower box(-counting) dimension 41 lower coarse multifractal spectrum 280 lower density 77, 84 lower limit lubrication theory 305 Mandelbrot, Benoit xxii, xxv Mandelbrot set 223, 223–227, 224, 230, 233, 235 mapping(s) 6, Markov partition 189 martingale 248, 311 mass distribution 11, 12, 277 construction by repeated subdivision 14–15, 15 and distribution of digits 151–152 and product rule 99, 101 uniform, on line segment 14 Index mass distribution principle 60, 60–61, 131, 200 maximum modulus theorem 231 maximum range 160 Maxwell’s equations 310 mean 21 mean-value theorem 10, 137, 190 measure(s) 11, 11–17 counting 13 Hausdorff 14, 27–30, 28, 53, 99, 113 Hausdorff dimension of 209, 288 invariant 208, 208–211 Lebesgue 13, 16, 28, 112, 192 multifractal 211, 277–296 n-dimensional Lebesgue 13, 17, 112, 143 net 36, 68 packing 50–53, 51, 88 probability 19 restriction of 14 self-similar 278, 279, 280, 286 on a set 11 σ -finite 95 tangent 89 r-mesh cube 42, 278 method of moments, for multifractal spectrum 280–281, 283 metric dimension 41 middle λ Cantor set 64 middle third Cantor set xvii, xviii as attractor 189 box(-counting) dimension 47 construction of xviii , 127 features xviii, 123 generalization of 63–64 Hausdorff dimension 34–35, 60, 61 in intersections 112 product 99, 100 and repellers 188, 189 and self-similarity 123, 124, 129 and tangents 87 Minkowski content 45 Minkowski(-Bouligand) dimension 46 modified box(-counting) dimension 49, 49–50 upper, and packing dimension 51–52 modified von Koch curve 132, 133–134 moment sum 280, 281 monotonicity of box(-)counting dimension 48 of dimension print 55 of Hausdorff dimension 32, 41 of packing dimension 51 Montel’s theorem 218–219, 221 Index Moser’s twist theorem 207 mountain skyline 271, 298 multifractal(s) 211, 277–296 coarse analysis 277, 278–283 fine analysis 277, 283–286 self-similar 286–296 multifractal spectrum 277–286, 283, 289, 292–294, 315 multifractal time 312, 313, 314 multifractional Brownian motion 271 multiple points 265, 275 multivariate normal variable(s) 267 natural fractals xxvi, 146, 147, 298–300 natural logarithms 10 Navier–Stokes equations 203, 307, 309 ‘nearest point mapping’ 181, 182 neighbourhood 4, 5, 45 see also (delta-)/(δ-)neighbourhood net measure 36, 68 neural networks 277 Newton’s method 237–241 non-linear Cantor set 136–137, 154 non-removable set 180, 181 non-singular transformation normal distribution 23, 260 normal family 218, 218–219 at a point 218 normal numbers 151 number theory 151–158 often 110 one-sided dimension 54 one-to-one correspondence one-to-one function onto function open ball open interval 4, 138 open set 5, 5, 32, 41, 187 open set condition 129, 130–134, 249 orbit 186, 189, 216, 228 orthogonal projection xxv, 34, 90–97, 91, 176, 177 packing dimension 50–53, 51, 284 and modified upper box dimension 51–52 packing measure 50–53, 51, 88 and Hausdorff measure 53 parallel body Parseval’s theorem 73 partial quotient 153 percolation 251–255, 252, 253, 254 335 perfect set 220, 242 period 216 period doubling 191–193 period-p orbit 216, 232, 235 period-p point 186, 191 periodic orbit 228, 232 periodic point 216, 221 phase transition 251 physical applications xxvi, 298–316 pinch point 232 plane cross section 202 plant growth 300 Poincar´e –Bendixson theorem 201, 202 Poincar´e section 202 point mass 13 pointwise convergence 10 Poisson’s equation 307 polynomials, Newton’s method for 237–241 population dynamics 186, 190, 193 porous medium, flow through 305 (s-)potential 70 potential theoretic methods 70–72, 92, 111, 248–249 power spectrum 171, 270 Prandtl number 203 pre-fractal(s) 126, 127 pre-image probability 18 conditional 19 probability density function 23 probability measure 18, 19 probability space 19 probability theory 17–24 product 99 Cartesian 4, 87, 99, 100 product formula 99–107 projection(s) 90–97, 91 of arbitrary sets 90–93 of arbitrary sets of integral dimension 95–97 of s-sets of integral dimension 93–95 projection theorems 90–93, 180 Pythagoras’s theorem 266 quadratic functions 223–235 quadratic polynomial, Newton’s method for 238–239 quasi-circle(s) 231, 235–237, 236 radio antennas 309–311 rainfall distribution 277 random Cantor set 245, 246, 246–250, 256 336 random fractal(s) 244–256 random function 259, 260, 270 random mid-point displacement method 260 random process 259 random variable 20 independence of 20–21 simple 21 random von Koch curve xxii, xxiii , 245, 251, 256 random walk 258–259 range, maximum 160 ratio of contraction 128 rational function 238 rectifiable curve 81, 81–84, 181–182 tangent 85–86 reflection regular point 78 regular set 78, 80–89, 93–94 examples 81 tangent 86–87 removable set 179, 179–181 repeller 187, 187–189, 215 repelling point 216, 233 closure of 221 reservoir level variations 160 residence measure 208–209, 277, 283 restriction of a measure 14 rigid motion 8, 109, 111 ring(s) of fractional dimension 182184 Răossler band 204, 204 rotation sample function 259, 262, 265 sample space 18 scalar multiple scaling property 29, 29 sections, parallel 105 sector, double 85 self-affine curve 166–169, 168, 312 construction of 166, 167 self-affine set 106–107, 107, 139, 139–145 as attractor 139, 140, 142, 144, 146 construction of 107, 142 self-similar fractal, with two different similarity ratios xxi self-similar measure 280, 286, 294 construction of 278, 279 multifractal spectrum 288, 293–296 self-similar multifractals xxiv, 286–296 self-similar set xxiv, 128–135, 128 similarity dimension xxiv Index see also middle third Cantor set; Sierpi´nski gasket; von Koch curve self-similarity xxii sensitive dependence on initial conditions 187, 194, 222 s-set 32, 69, 76 0-set 77 1-set 80–84, 86–87, 93–94 tangent to 84–88, 85 set theory 3–6 share prices 160, 258, 277, 311–315 Siegel disc 232 Sierpi´nski dipole 310–311 Sierpi´nski gasket or triangle xx, xx, 129, 132, 256 (sigma-)/σ -finite measure 95 similarity 7, 8, 29, 111, 128 similarity dimension xxiv simple closed curve 230, 235 simple function 16 simple random variable 21 simulation 299 singular value(s) 142 singular value function 143 singularity set 306 singularity spectrum 284 small divisor theory 205–208 smooth manifold 41 smooth set 32 snowflake curve xix solenoid 198–201, 199, 200 solid square, dimension print 56 solution curves 201 stability 41, 48 countable 32, 41, 55 stable point 191 stable process 271, 271–273, 275 stable set 216 stable symmetric process 272, 273 stationary increments 259, 267, 271 statistically self-affine set 262, 268, 311 statistically self-similar set 244, 246, 251, 262, 268 stock market prices 160, 258, 277, 311–315 strange attractor 186, 205 stretching and folding or cutting transformations 193–197, 210 strong law of large numbers 24, 152 subgroup 182, 183 submanifold 29, 48 submultiplicative sequence 143 subring 183, 183–184 examples 183 Index subset of finite measure 68–70 sundial, digital 96–97, 97 superattractive point 238 support of a measure 12 supremum surface, convex 182 surjection symbolic dynamics 189, 212 tangent xxv–xxvi, 84–89, 85, 86 tangent measure 89 tangent plane 182 tendril 232 tends to tends to infinity tent map 188, 188–189, 188 tent map repeller 188, 189, 190 thermal convection 202 topological dimension xxv torus 198–201 totally disconnected set(s) 6, 33, 81, 84, 136, 228, 255 trading time 312 trail 258, 262, 264 trajectories 201–202 transformation(s) 6, 7–8 effects on a set group of 110, 111 linear stretching and folding or cutting 193–198, 210 translations 7–8 tree antenna 310, 311 trial 18 turbulence 307–309 fractally homogeneous 309 homogeneous 308 isotropic 307 turbulent flow 307 twist map 207 337 ubiquity theorem 118 uncountable set uniform Cantor set 63–64, 64, 103 uniform convergence 10 uniform distribution 23 union of sets unstable point 191 upper box(-counting) dimension 41 upper coarse multifractal spectrum 280 upper density of s-set 77 upper limit variance 22 vector sum of sets ‘Venetian blind’ construction, iterated 95–96, 96, 180 viscous fingering 277, 300, 304–305 Vitushkin’s conjecture 179–181 volume 14, 45–46 scaling of 29 von Koch curve xviii–xx, xix construction of xix , 127–128 dimension of xxiii features xx, 35, 123 modified 132, 133–134 random xxii, xxiii , 245, 251, 256 as self-similar set 123, 129 von Koch dipole 310 von Koch snowflake xix vortex tubes 309 weak solution 307 weather prediction 204 Weierstrass function xxiii, 160, 162–166, 164–165 random 270 (α-)well approximable number(s) 155, 155–157, 205, 207 Wiener process 258, 259 ... 1 87 189 193 198 20 1 20 5 20 8 21 1 21 2 21 5 21 5 22 3 2 27 23 5 2 37 24 1 24 2 viii Contents Chapter 15 Random fractals 15.1 15 .2 15.3 A random Cantor set Fractal. .. Chapter 18 Physical applications 18.1 18 .2 18.3 18.4 18.5 18 .6 24 6 25 1 25 5 25 6 27 8 28 3 28 6 29 6 29 6 29 8 300 3 06 3 07 309 311 315 3 16 References ... 0- 4 70 -84 861 -8 (HB); 0- 4 70 -848 62 - 6 (PB) FRACTAL GEOMETRY Mathematical Foundations and Applications Second Edition Kenneth Falconer University of St Andrews, UK Copyright  20 0 3 John Wiley & Sons