Preface p. ix
Preface to the second edition p. xiii
Course suggestions p. xv
Introduction p. xvii
Notes and references p. xxvii
Foundations p. 1
Mathematical background p. 3
Basic set theory p. 3
Functions and limits p. 6
Measures and mass distributions p. 11
Notes on probability theory p. 17
Notes and references p. 24
Exercises p. 25
Hausdorff measure and dimension p. 27
Hausdorff measure p. 27
Hausdorff dimension p. 31
Calculation of Hausdorff dimension--simple examples p. 34
Equivalent definitions of Hausdorff dimension p. 35
Finer definitions of dimension p. 36
Notes and references p. 37
Exercises p. 37
Alternative definitions of dimension p. 39
Box-counting dimensions p. 41
Properties and problems of box-counting dimension p. 47
Modified box-counting dimensions p. 49
Packing measures and dimensions p. 50
Some other definitions of dimension p. 53
Notes and references p. 57
Exercises p. 57
Techniques for calculating dimensions p. 59
Basic methods p. 59
Subsets of finite measure p. 68
Potential theoretic methods p. 70
Fourier transform methods p. 73
Notes and references p. 74
Exercises p. 74
Local structure of fractals p. 76
Densities p. 76
Structure of 1-sets p. 80
Tangents to s-sets p. 84
Notes and references p. 89
Exercises p. 89
Projections of fractals p. 90
Projections of arbitrary sets p. 90
Projections of s-sets of integral dimension p. 93
Projections of arbitrary sets of integral dimension p. 95
Notes and references p. 97
Exercises p. 97
Products of fractals p. 99
Product formulae p. 99
Notes and references p. 107
Exercises p. 107
Intersections of fractals p. 109
Intersection formulae for fractals p. 110
Sets with large intersection p. 113
Notes and references p. 118
Exercises p. 119
Applications and Examples p. 121
Iterated function systems--self-similar and self-affine sets p. 123
Iterated function systems p. 123
Dimensions of self-similar sets p. 128
Some variations p. 135
Self-affine sets p. 139
Applications to encoding images p. 145
Notes and references p. 148
Exercises p. 149
Examples from number theory p. 151
Distribution of digits of numbers p. 151
Continued fractions p. 153
Diophantine approximation p. 154
Notes and references p. 158
Exercises p. 158
Graphs of functions p. 160
Dimensions of graphs p. 160
Autocorrelation of fractal functions p. 169
Notes and references p. 173
Exercises p. 173
Examples from pure mathematics p. 176
Duality and the Kakeya problem p. 176
Vitushkin's conjecture p. 179
Convex functions p. 181
Groups and rings of fractional dimension p. 182
Notes and references p. 184
Exercises p. 185
Dynamical systems p. 186
Repellers and iterated function systems p. 187
The logistic map p. 189
Stretching and folding transformations p. 193
The solenoid p. 198
Continuous dynamical systems p. 201
Small divisor theory p. 205
Liapounov exponents and entropies p. 208
Notes and references p. 211
Exercises p. 212
Iteration of complex functions--Julia sets p. 215
General theory of Julia sets p. 215
Quadratic functions--the Mandelbrot set p. 223
Julia sets of quadratic functions p. 227
Characterization of quasi-circles by dimension p. 235
Newton's method for solving polynomial equations p. 237
Notes and references p. 241
Exercises p. 242
Random fractals p. 244
A random Cantor set p. 246
Fractal percolation p. 251
Notes and references p. 255
Exercises p. 256
Brownian motion and Brownian surfaces p. 258
Brownian motion p. 258
Fractional Brownian motion p. 267
Levy stable processes p. 271
Fractional Brownian surfaces p. 273
Notes and references p. 275
Exercises p. 276
Multifractal measures p. 277
Coarse multifractal analysis p. 278
Fine multifractal analysis p. 283
Self-similar multifractals p. 286
Notes and references p. 296
Exercises p. 296
Physical applications p. 298
Fractal growth p. 300
Singularities of electrostatic and gravitational potentials p. 306
Fluid dynamics and turbulence p. 307
Fractal antennas p. 309
Fractals in finance p. 311
Notes and references p. 315
Exercises p. 316
References p. 317
Index p. 329