Curves and Fractal Dimension
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1 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
2 Contents Foreword, by Michel Mendes France Introduction v vii Part I. Sets of Null Measure on the Line 1. Perfect Sets and Their Measure Duality set measure Closed sets and contiguous intervals Perfect sets Binary trees and the power of perfect sets Symmetrical perfect sets Tree representation of perfect sets Bibliographical notes Covers and Dimension What is a null measure? Hierarchy of sets of null measure Cantor-Minkowski measure Space Alling and the order of growth Orders of growth and dimension Equivalent definitions of the dimension Examples of Computing the dimension Some properties of the dimension Upper and lower dimensions Bibliographical notes Contiguous Intervals and Dimension Borel's logarithmic rarefaction Index of Besicovitch-Taylor Equivalent Orders of growth The contiguous intervals and the fractal dimension Algorithms to compute the dimension Bibliographical notes 40
3 Contents XI Part II. Rectifiable Curves 4. What Is a Curve? Some types of sets in the plane Velocities, trajectories The definition of a curve Bibliographical notes Polygonal Curves and Length Rectifiability Hausdorff distance Polygonal approximations The length of a curve Two distinct notions Measuring the length by compass Bibliographical notes Parameterized Curves, Support of a Measure Parameterization by arc length Image measure Length by instantaneous velocity The devil staircase Length by the average of local velocity Bibliographical notes Local Geometry of Rectifiable Curves Tangent, cone, convex hulls Relations between local properties Counterexamples Tangent almost everywhere Local length, almost everywhere Rectifiability revisited Bibliographical notes Length, by Intersections with Straight Lines Intersections, projections The measure of families of straight lines Family of lines intersecting a set The case of convex sets Length by secant lines The length by projections Application: practical computation of length The length by random intersections Buffon needle 101
4 XII Contents 8.10 Bibliographical notes The Length by the Area of Centered Balls Minkowski sausage Length by the area of sausages Convergence of the algorithm of the sausages Reduction of balls to parallel segments Bibliographical notes 114 Part III. Nonrectifiable Curves 10. Curves of Infinite Length What is infinite length? Two examples Dimension Some examples of dimensions of curves Classical Covers: balls and boxes Covers by figures of any kind Covering curves by crosses Bibliographical notes Fractal Curves What is a fractal curve? A fractal curve is nowhere rectifiable Diameter, size Characterization of a fractal curve Graphs of Nondifferentiable Functions Curves parameterized by the abscissa Size of local arcs Variation of a function Practal dimension of a graph Holder exponent Functions defined by series Weierstrass function Fractal dimension and the structure function Functions constructed by diagonal affinities Invariance under change of scale The Weierstrass-Mandelbrot function The spectrum of invariant functions Computing the dimensions of the graphs Bibliographical notes 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.](/docs-images/87/95683102/images/5-0.jpg "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.")
5 Contents XIII 13.1 Similarities Self-similar structure Generator Self-similar structure on [0,1] Parameterization of the generator The limit curve T Simplicity criterion Similarity and dimension exponent Examples The natural parameterization The algorithm of local sizes Bibliographical notes Deviation, and Expansive Curves Introducing new notions Deviation of a set Constant deviation along a curve Definition of an expansive curve Expansivity criterion Expansivity and self-similarity How to construct an expansive curve Bibliographical notes The Constant-Deviation Variable-Step Algorithm A unified analysis of expansive curves The covering index Convex hulls and Minkowski sausages A theorem on the dimension: the discrete form Applications Statistical self-similarity Curves of uniform deviation Applications The dimension of a curve Bibliographical notes Scanning a Curve with Straight Lines Directional dimension Comparing the dimensions Examples and applications Coordinate Systems Intersections by straight lines Essential upper bound Uniform intersections Intersection with an average curve Bibliographical notes 258
6 XIV Contents 17. Lateral Dimension of a Curve Semisausages Other expressions of the lateral dimensions Possible values of the lateral dimension Examples The inverse Minkowski Operation Bibliographical notes Dimensional Homogeneity Local structures of some curves Local dimension The packing dimension Possible values of the packing dimension The <7-stabilization 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
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