Fractal geometry : mathematical foundations and applications / Kenneth Falconer.

Format:

Book

Author/Creator:

Falconer, K. J., 1952-

Contributor:

Sabin W. Colton, Jr., Memorial Fund.

Language:

English

Subjects (All):

Fractals.

Fractals--Problems, exercises, etc.

Genre:

Problems and exercises.

Physical Description:

xxvii, 337 pages : illustrations ; 24 cm

Edition:

Second edition.

Place of Publication:

Chichester ; Hoboken, NJ : Wiley, 2003.

Summary:

Since its original publication in 1990, Kenneth Falconer's "Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals. It introduces the general mathematical theory and applications of fractals in a way that is accessible to students from a wide range of disciplines. This new edition has been extensively revised and updated. It features much new material, many additional exercises, notes and references, and an extended bibliography that reflects the development of the subject since the first edition.

· Provides a comprehensive and accessible introduction to the mathematical theory and applications of fractals.

· Each topic is carefully explained and illustrated by examples and figures.

· Includes all necessary mathematical background material.

· Includes notes and references to enable the reader to pursue individual topics.

· Features a wide selection of exercises, enabling the reader to develop their understanding of the theory.

· Supported by a Web site featuring solutions to exercises, and additional material for students and lecturers.

"Fractal Geometry: Mathematical Foundations and Applications is aimed at undergraduate and graduate students studying courses in fractal geometry. The book also provides an excellent source of reference for researchers who encounter fractals in mathematics, physics, engineering, and the applied sciences.

"Also by Kenneth Falconer and available from Wiley:

"Techniques in Fractal Geometry

ISBN 0-471-95724-0

Contents:

Chapter 1 Mathematical background 3

1.1 Basic set theory 3

1.2 Functions and limits 6

1.3 Measures and mass distributions 11

1.4 Notes on probability theory 17

Chapter 2 Hausdorff measure and dimension 27

2.2 Hausdorff dimension 31

2.3 Calculation of Hausdorff dimension

simple examples 34

2.4 Equivalent definitions of Hausdorff dimension 35

2.5 Finer definitions of dimension 36

Chapter 3 Alternative definitions of dimension 39

3.1 Box-counting dimensions 41

3.2 Properties and problems of box-counting dimension 47

3.3 Modified box-counting dimensions 49

3.4 Packing measures and dimensions 50

Chapter 4 Techniques for calculating dimensions 59

4.2 Subsets of finite measure 68

4.3 Potential theoretic methods 70

4.4 Fourier transform methods 73

Chapter 5 Local structure of fractals 76

5.1 Densities 76

5.2 Structure of 1-sets 80

5.3 Tangents to s-sets 84

Chapter 6 Projections of fractals 90

6.1 Projections of arbitrary sets 90

6.2 Projections of s-sets of integral dimension 93

6.3 Projections of arbitrary sets of integral dimension 95

Chapter 7 Products of fractals 99

Chapter 8 Intersections of fractals 109

8.2 Sets with large intersection 113

Part II Applications and Examples 121

Chapter 9 Iterated function systems

self-similar and self-affine sets 123

9.1 Iterated function systems 123

9.2 Dimensions of self-similar sets 128

9.3 Some variations 135

9.4 Self-affine sets 139

9.5 Applications to encoding images 145

Chapter 10 Examples from number theory 151

10.1 Distribution of digits of numbers 151

10.2 Continued fractions 153

10.3 Diophantine approximation 154

Chapter 11 Graphs of functions 160

11.1 Dimensions of graphs 160

11.2 Autocorrelation of fractal functions 169

Chapter 12 Examples from pure mathematics 176

12.1 Duality and the Kakeya problem 176

12.2 Vitushkin's conjecture 179

12.3 Convex functions 181

12.4 Groups and rings of fractional dimension 182

Chapter 13 Dynamical systems 186

13.1 Repellers and iterated function systems 187

13.2 The logistic map 189

13.3 Stretching and folding transformations 193

13.4 The solenoid 198

13.5 Continuous dynamical systems 201

13.6 Small divisor theory 205

13.7 Liapounov exponents and entropies 208

Chapter 14 Iteration of complex functions

Julia sets 215

14.1 General theory of Julia sets 215

14.2 Quadratic functions

the Mandelbrot set 223

14.3 Julia sets of quadratic functions 227

14.4 Characterization of quasi-circles by dimension 235

14.5 Newton's method for solving polynomial equations 237

Chapter 15 Random fractals 244

15.1 A random Cantor set 246

15.2 Fractal percolation 251

Chapter 16 Brownian motion and Brownian surfaces 258

16.1 Brownian motion 258

16.2 Fractional Brownian motion 267

16.3 Levy stable processes 271

16.4 Fractional Brownian surfaces 273

Chapter 17 Multifractal measures 277

17.1 Coarse multifractal analysis 278

17.2 Fine multifractal analysis 283

17.3 Self-similar multifractals 286

Chapter 18 Physical applications 298

18.1 Fractal growth 300

18.2 Singularities of electrostatic and gravitational potentials 306

18.3 Fluid dynamics and turbulence 307

18.4 Fractal antennas 309

18.5 Fractals in finance 311.

Notes:

Previous ed.: 1990.

Includes bibliographical references (pages 317-328) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Sabin W. Colton, Jr., Memorial Fund.

ISBN:

0470848618

0470848626

OCLC:

52486585