Fractal Geometry

Preface to the first edition ix Preface to the second edition xiii Preface to the third edition xv Course suggestions xvii Introduction xix PART I FOUNDATIONS 1 1 Mathematical background 3 1.1 Basic set theory 3 1.2 Functions and limits 7 1.3 Measures and mass distributions 11 1.4 Notes on probability theory 17 1.5 Notes and references 24 Exercises 24 2 Box-counting dimension 27 2.1 Box-counting dimensions 27 2.2 Properties and problems of box-counting dimension 34 *2.3 Modified box-counting dimensions 38 2.4 Some other definitions of dimension 40 2.5 Notes and references 41 Exercises 42 3 Hausdorff and packing measures and dimensions 44 3.1 Hausdorff measure 44 3.2 Hausdorff dimension 47 3.3 Calculation of Hausdorff dimension – simple examples 51 3.4 Equivalent definitions of Hausdorff dimension 53 *3.5 Packing measure and dimensions 54 *3.6 Finer definitions of dimension 57 *3.7 Dimension prints 58 *3.8 Porosity 60 3.9 Notes and references 63 Exercises 64 4 Techniques for calculating dimensions 66 4.1 Basic methods 66 4.2 Subsets of finite measure 75 4.3 Potential theoretic methods 77 *4.4 Fourier transform methods 80 4.5 Notes and references 81 Exercises 81 5 Local structure of fractals 83 5.1 Densities 84 5.2 Structure of 1-sets 87 5.3 Tangents to s-sets 92 5.4 Notes and references 96 Exercises 96 6 Projections of fractals 98 6.1 Projections of arbitrary sets 98 6.2 Projections of s-sets of integral dimension 101 6.3 Projections of arbitrary sets of integral dimension 103 6.4 Notes and references 105 Exercises 106 7 Products of fractals 108 7.1 Product formulae 108 7.2 Notes and references 116 Exercises 116 8 Intersections of fractals 118 8.1 Intersection formulae for fractals 119 *8.2 Sets with large intersection 122 8.3 Notes and references 128 Exercises 128 PART II APPLICATIONS AND EXAMPLES 131 9 Iterated function systems – self-similar and self-affine sets 133 9.1 Iterated function systems 133 9.2 Dimensions of self-similar sets 139 CONTENTS vii 9.3 Some variations 143 9.4 Self-affine sets 149 9.5 Applications to encoding images 155 *9.6 Zeta functions and complex dimensions 158 9.7 Notes and references 167 Exercises 167 10 Examples from number theory 169 10.1 Distribution of digits of numbers 169 10.2 Continued fractions 171 10.3 Diophantine approximation 172 10.4 Notes and references 176 Exercises 176 11 Graphs of functions 178 11.1 Dimensions of graphs 178 *11.2 Autocorrelation of fractal functions 188 11.3 Notes and references 192 Exercises 192 12 Examples from pure mathematics 195 12.1 Duality and the Kakeya problem 195 12.2 Vitushkin’s conjecture 198 12.3 Convex functions 200 12.4 Fractal groups and rings 201 12.5 Notes and references 204 Exercises 204 13 Dynamical systems 206 13.1 Repellers and iterated function systems 208 13.2 The logistic map 209 13.3 Stretching and folding transformations 213 13.4 The solenoid 217 13.5 Continuous dynamical systems 220 *13.6 Small divisor theory 225 *13.7 Lyapunov exponents and entropies 228 13.8 Notes and references 231 Exercises 232 14 Iteration of complex functions – Julia sets and the Mandelbrot set 235 14.1 General theory of Julia sets 235 14.2 Quadratic functions – the Mandelbrot set 243 14.3 Julia sets of quadratic functions 248 14.4 Characterisation of quasi-circles by dimension 256 14.5 Newton’s method for solving polynomial equations 258 14.6 Notes and references 262 Exercises 262 15 Random fractals 265 15.1 A random Cantor set 266 15.2 Fractal percolation 272 15.3 Notes and references 277 Exercises 277 16 Brownian motion and Brownian surfaces 279 16.1 Brownian motion in ℝ 279 16.2 Brownian motion in ℝn 285 16.3 Fractional Brownian motion 289 16.4 Fractional Brownian surfaces 294 16.5 Lévy stable processes 296 16.6 Notes and references 299 Exercises 299 17 Multifractal measures 301 17.1 Coarse multifractal analysis 302 17.2 Fine multifractal analysis 307 17.3 Self-similar multifractals 310 17.4 Notes and references 320 Exercises 320 18 Physical applications 323 18.1 Fractal fingering 325 18.2 Singularities of electrostatic and gravitational potentials 330 18.3 Fluid dynamics and turbulence 332 18.4 Fractal antennas 334 18.5 Fractals in finance 336 18.6 Notes and references 340 Exercises 341 References 342 Index 357