Claude Tricot

Curves and Fractal Dimension

With a Foreword by Michel Mendes France

With 163 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest

Contents

Foreword, by Michel Mendes France v

Introduction vii

Part I. Sets of Null Measure on the Line

1. Perfect Sets and Their Measure 1

1.1 Duality set—measure 1 1.2 Closed sets and contiguous intervals 2 1.3 Perfect sets 4 1.4 Binary trees and the power of perfect sets 5 1.5 Symmetrical perfect sets 8 1.6 Tree representation of perfect sets 9 1.7 Bibliographical notes 12

2. Covers and Dimension 13 2.1 What is a null measure? 13 2.2 Hierarchy of sets of null measure 15 2.3 Cantor-Minkowski measure 16 2.4 Space Alling and the order of growth 19 2.5 Orders of growth and dimension 20 2.6 Equivalent definitions of the dimension 23 2.7 Examples of Computing the dimension 25 2.8 Some properties of the dimension 26 2.9 Upper and lower dimensions 27 2.10 Bibliographical notes 29

3. Contiguous Intervals and Dimension 33

3.1 Borel's logarithmic rarefaction 33 3.2 Index of Besicovitch-Taylor 34 3.3 Equivalent Orders of growth 34 3.4 The contiguous intervals and the fractal dimension 36 3.5 Algorithms to compute the dimension 38 3.6 Bibliographical notes 40

Contents XI

Part II. Rectifiable Curves

4. What Is a Curve? 43 4.1 Some types of sets in the plane 43 4.2 Velocities, trajectories 44 4.3 The definition of a curve 45 4.4 Bibliographical notes 46

5. Polygonal Curves and Length 47 5.1 Rectifiability 47 5.2 Hausdorff distance 47 5.3 Polygonal approximations •.. 50 5.4 The length of a curve 51 5.5 Two distinct notions 54 5.6 Measuring the length by compass 56 5.7 Bibliographical notes 57

6. Parameterized Curves, Support of a Measure 59 6.1 Parameterization by arc length 59 6.2 Image measure 60 6.3 Length by instantaneous velocity 60 6.4 The devil staircase 62 6.5 Length by the average of local velocity 67 6.6 Bibliographical notes 70

7. Local Geometry of Rectifiable Curves 71 7.1 Tangent, cone, convex hulls 71 7.2 Relations between local properties 73 7.3 Counterexamples 75 7.4 Tangent almost everywhere 78 7.5 Local length, almost everywhere 80 7.6 Rectifiability revisited 81 7.7 Bibliographical notes 82

8. Length, by Intersections with Straight Lines 85 8.1 Intersections, projections 85 8.2 The measure of families of straight lines 86 8.3 Family of lines intersecting a set 89 8.4 The case of convex sets 91 8.5 Length by secant lines 93 8.6 The length by projections 97 8.7 Application: practical computation of length 98 8.8 The length by random intersections 100 8.9 Buffon needle 101

XII Contents

8.10 Bibliographical notes 102

9. The Length by the Area of Centered Balls 105 9.1 Minkowski sausage 105 9.2 Length by the area of sausages 106 9.3 Convergence of the algorithm of the sausages 110 9.4 Reduction of balls to parallel segments 112 9.5 Bibliographical notes 114

Part III. Nonrectifiable Curves

10. Curves of Infinite Length 115 10.1 What is infinite length? 115 10.2 Two examples 116 10.3 Dimension 119 10.4 Some examples of dimensions of curves 120 10.5 Classical Covers: balls and boxes 123 10.6 Covers by figures of any kind 128 10.7 Covering curves by crosses 130 10.8 Bibliographical notes 133

11. Fractal Curves 135 11.1 What is a fractal curve? 135 11.2 A fractal curve is nowhere rectifiable 137 11.3 Diameter, size 139 11.4 Characterization of a fractal curve 141

12. Graphs of Nondifferentiable Functions 143 12.1 Curves parameterized by the abscissa 143 12.3 Size of local arcs 144 12.3 Variation of a function 145 12.4 Practal dimension of a graph 148 12.5 Holder exponent 150 12.6 Functions defined by series 152 12.7 Weierstrass function 154 12.8 Fractal dimension and the structure function 157 12.9 Functions constructed by diagonal affinities 160 12.10 Invariance under change of scale 163 12.11 The Weierstrass-Mandelbrot function 166 12.12 The spectrum of invariant functions 168 12.13 Computing the dimensions of the graphs 170 12.14 Bibliographical notes 174

13. Curves Constructed by Similarities 177

Contents XIII

13.1 Similarities 177 13.2 Self-similar structure 179 13.3 Generator 180 13.4 Self-similar structure on [0,1] 182 13.5 Parameterization of the generator 183 13.6 The limit curve T 185 13.7 Simplicity criterion 187 13.8 Similarity and dimension exponent 190 13.9 Examples 192 13.10 The natural parameterization 195 13.11 The algorithm of local sizes 199 13.12 Bibliographical notes 201

14. Deviation, and Expansive Curves 203 14.1 Introducing new notions 203 14.2 Deviation of a set 203 14.3 Constant deviation along a curve 206 14.4 Definition of an expansive curve 208 14.5 Expansivity criterion 209 14.6 Expansivity and self-similarity 213 14.7 How to construct an expansive curve 214 14.8 Bibliographical notes 220

15. The Constant-Deviation Variable-Step Algorithm 221 15.1 A unified analysis of expansive curves 221 15.2 The covering index 222 15.3 Convex hulls and Minkowski sausages 223 15.4 A theorem on the dimension: the discrete form 225 15.5 Applications 227 15.6 Statistical self-similarity 232 15.7 Curves of uniform deviation 234 15.8 Applications 236 15.9 The dimension of a curve 239 15.10 Bibliographical notes 242

16. Scanning a Curve with Straight Lines 243 16.1 Directional dimension 243 16.2 Comparing the dimensions 245 16.3 Examples and applications 246 16.4 Coordinate Systems 247 16.5 Intersections by straight lines 251 16.6 Essential upper bound 253 16.7 Uniform intersections 255 16.8 Intersection with an average curve 256 16.9 Bibliographical notes 258

XIV Contents

17. Lateral Dimension of a Curve 259 17.1 Semisausages 259 17.2 Other expressions of the lateral dimensions 260 17.3 Possible values of the lateral dimension 263 17.4 Examples 264 17.5 The inverse Minkowski Operation 268 17.6 Bibliographical notes 271

18. Dimensional Homogeneity 273 18.1 Local structures of some curves 273 18.2 Local dimension 274 18.3 The packing dimension 277 18.4 Possible values of the packing dimension 279 18.5 The <7-stabilization 282 18.6 Bibliographical notes 283

Part IV. Annexes, References and Index

A. Upper Limit and Lower Limit 285 A.l Convergence 285 A.2 Nonconvergent sequences 287 A.3 Nonconvergent Functions 288 A.4 Limits of the ratio log / ( e ) / \ogg(e) 289 A.5 Some applications 291

B. Two Covering Lemmas 293

B.l Vitali's lemma 293 B.2 Covers by homothetic convex sets 296

C. Convex Sets in the Plane 301

C.l Convexity 301 C.2 Size of a convex set 302 C.3 Breadth of a convex set 305 C.4 Area of a convex set 310 C.5 Convex hüll 310 C.6 Perimeter of the convex hüll 312 C.7 Area of the convex hüll of a curve 313

References 315

Index 318