Curves and Fractal Dimension

I. Sets of Null Measure on the Line.- 1. Perfect Sets and Their Measure.- 1.1 Duality set—measure.- 1.2 Closed sets and contiguous intervals.- 1.3 Perfect sets.- 1.4 Binary trees and the power of perfect sets.- 1.5 Symmetrical perfect sets.- 1.6 Tree representation of perfect sets.- 1.7 Bibliographical notes.- 2. Covers and Dimension.- 2.1 What is a null measure?.- 2.2 Hierarchy of sets of null measure.- 2.3 Cantor-Minkowski measure.- 2.4 Space filling and the order of growth.- 2.5 Orders of growth and dimension.- 2.6 Equivalent definitions of the dimension.- 2.7 Examples of computing the dimension.- 2.8 Some properties of the dimension.- 2.9 Upper and lower dimensions.- 2.10 Bibliographical notes.- 3. Contiguous Intervals and Dimension.- 3.1 Borel’s logarithmic rarefaction.- 3.2 Index of Besicovitch-Taylor.- 3.3 Equivalent orders of growth.- 3.4 The contiguous intervals and the fractal dimension.- 3.5 Algorithms to compute the dimension.- 3.6 Bibliographical notes.- II. Rectifiable Curves.- 4. What Is a Curve?.- 4.1 Some types of sets in the plane.- 4.2 Velocities, trajectories.- 4.3 The definition of a curve.- 4.4 Bibliographical notes.- 5. Polygonal Curves and Length.- 5.1 Rectifiability.- 5.2 Hausdorff distance.- 5.3 Polygonal approximations.- 5.4 The length of a curve.- 5.5 Two distinct notions.- 5.6 Measuring the length by compass.- 5.7 Bibliographical notes.- 6. Parameterized Curves, Support of a Measure.- 6.1 Parameterization by arc length.- 6.2 Image measure.- 6.3 Length by instantaneous velocity.- 6.4 The devil staircase.- 6.5 Length by the average of local velocity.- 6.6 Bibliographical notes.- 7. Local Geometry of Rectifiable Curves.- 7.1 Tangent, cone, convex hulls.- 7.2 Relations between local properties.- 7.3 Counterexamples.- 7.4 Tangent almost everywhere.- 7.5 Local length, almost everywhere.- 7.6 Rectifiability revisited.- 7.7 Bibliographical notes.- 8. Length, by Intersections with Straight Lines.- 8.1 Intersections, projections.- 8.2 The measure of families of straight lines.- 8.3 Family of lines intersecting a set.- 8.4 The case of convex sets.- 8.5 Length by secant lines.- 8.6 The length by projections.- 8.7 Application: practical computation of length.- 8.8 The length by random intersections.- 8.9 Buffon needle.- 8.10 Bibliographical notes.- 9. The Length by the Area of Centered Balls.- 9.1 Minkowski sausage.- 9.2 Length by the area of sausages.- 9.3 Convergence of the algorithm of the sausages.- 9.4 Reduction of balls to parallel segments.- 9.5 Bibliographical notes.- III. Nonrectifiable Curves.- 10. Curves of Infinite Length.- 10.1 What is infinite length?.- 10.2 Two examples.- 10.3 Dimension.- 10.4 Some examples of dimensions of curves.- 10.5 Classical covers: balls and boxes.- 10.6 Covers by figures of any kind.- 10.7 Covering curves by crosses.- 10.8 Bibliographical notes.- 11. Fractal Curves.- 11.1 What is a fractal curve?.- 11.2 A fractal curve is nowhere rectifiable.- 11.3 Diameter, size.- 11.4 Characterization of a fractal curve.- 12. Graphs of Nondifferentiable Functions.- 12.1 Curves parameterized by the abscissa.- 12.3 Size of local arcs.- 12.3 Variation of a function.- 12.4 Fractal dimension of a graph.- 12.5 Hölder exponent.- 12.6 Functions defined by series.- 12.7 Weierstrass function.- 12.8 Fractal dimension and the structure function.- 12.9 Functions constructed by diagonal affinities.- 12.10 Invariance under change of scale.- 12.11 The Weierstrass-Mandelbrot function.- 12.12 The spectrum of invariant functions.- 12.13 Computing the dimensions of the graphs.- 12.14 Bibliographical notes.- 13. Curves Constructed by Similarities.- 13.1 Similarities.- 13.2 Self-similar structure.- 13.3 Generator.- 13.4 Self-similar structure on [0,1].- 13.5 Parameterization of the generator.- 13.6 The limit curve ?.- 13.7 Simplicity criterion.- 13.8 Similarity and dimension exponent.- 13.9 Examples.- 13.10 The natural parameterization.- 13.11 The algorithm of local sizes.- 13.12 Bibliographical notes.- 14. Deviation, and Expansive Curves.- 14.1 Introducing new notions.- 14.2 Deviation of a set.- 14.3 Constant deviation along a curve.- 14.4 Definition of an expansive curve.- 14.5 Expansivity criterion.- 14.6 Expansivity and self-similarity.- 14.7 How to construct an expansive curve.- 14.8 Bibliographical notes.- 15. The Constant-Deviation Variable-Step Algorithm.- 15.1 A unified analysis of expansive curves.- 15.2 The covering index.- 15.3 Convex hulls and Minkowski sausages.- 15.4 A theorem on the dimension: the discrete form.- 15.5 Applications.- 15.6 Statistical self-similarity.- 15.7 Curves of uniform deviation.- 15.8 Applications.- 15.9 The dimension of a curve.- 15.10 Bibliographical notes.- 16. Scanning a Curve with Straight Lines.- 16.1 Directional dimension.- 16.2 Comparing the dimensions.- 16.3 Examples and applications.- 16.4 Coordinate systems.- 16.5 Intersections by straight lines.- 16.6 Essential upper bound.- 16.7 Uniform intersections.- 16.8 Intersection with an average curve.- 16.9 Bibliographical notes.- 17. Lateral Dimension of a Curve.- 17.1 Semisausages.- 17.2 Other expressions of the lateral dimensions.- 17.3 Possible values of the lateral dimension.- 17.4 Examples.- 17.5 The inverse Minkowski operation.- 17.6 Bibliographical notes.- 18. Dimensional Homogeneity.- 18.1 Local structures of some curves.- 18.2 Local dimension.- 18.3 The packing dimension.- 18.4 Possible values of the packing dimension.- 18.5 The ?-stabilization.- 18.6 Bibliographical notes.- IV. Annexes, References and Index.- A. Upper Limit and Lower Limit.- A.1 Convergence.- A.2 Nonconvergent sequences.- A.3 Nonconvergent functions.- A.5 Some applications.- B. Two Covering Lemmas.- B.1 Vitali’s lemma.- B.2 Covers by homothetic convex sets.- C. Convex Sets in the Plane.- C.1 Convexity.- C.2 Size of a convex set.- C.3 Breadth of a convex set.- C.4 Area of a convex set.- C.5 Convex hull.- C.6 Perimeter of the convex hull.- C.7 Area of the convex hull of a curve.- References.