# Fractal Geometry: Mathematical Foundations and Applications, 3rd Edition

## Books

The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions.

Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals.  The book introduces and develops the general theory and applications of fractals in a way that is accessible to students and researchers from a wide range of disciplines.

Fractal Geometry: Mathematical Foundations and Applications is an excellent course book for undergraduate and graduate students studying fractal geometry, with suggestions for material appropriate for a first course indicated. The book also provides an invaluable foundation and reference for researchers who encounter fractals not only in mathematics but also in other areas across physics, engineering and the applied sciences.

• Provides a comprehensive and accessible introduction to the mathematical theory and     applications of fractals
• Carefully explains each topic using illustrative examples and diagrams
• Includes the necessary mathematical background material, along with notes and references to enable the reader to pursue individual topics
• Features a wide range of exercises, enabling readers to consolidate their understanding
• Supported by a website with solutions to exercises and additional material http://www.wileyeurope.com/fractal
Leads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)

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

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