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)
Author(s): Kenneth Falconer
Edition: Third Edition
Publisher: Wiley
Year: 2014
Language: English
Pages: 368
Cover
Title Page
Copyright
Contents
Preface to the first edition
Preface to the second edition
Preface to the third edition
Course suggestions
Introduction
Part I Foundations
Chapter 1 Mathematical background
1.1 Basic set theory
1.2 Functions and limits
1.3 Measures and mass distributions
1.4 Notes on probability theory
1.5 Notes and references
Exercises
Chapter 2 Box-counting dimension
2.1 Box-counting dimensions
2.2 Properties and problems of box-counting dimension
2.3 Modified box-counting dimensions
2.4 Some other definitions of dimension
2.5 Notes and references
Exercises
Chapter 3 Hausdorff and packing measures and dimensions
3.1 Hausdorff measure
3.2 Hausdorff dimension
3.3 Calculation of Hausdorff dimension-simple examples
3.4 Equivalent definitions of Hausdorff dimension
3.5 Packing measure and dimensions
3.6 Finer definitions of dimension
3.7 Dimension prints
3.8 Porosity
3.9 Notes and references
Exercises
Chapter 4 Techniques for calculating dimensions
4.1 Basic methods
4.2 Subsets of finite measure
4.3 Potential theoretic methods
4.4 Fourier transform methods
4.5 Notes and references
Exercises
Chapter 5 Local structure of fractals
5.1 Densities
5.2 Structure of 1-sets
5.3 Tangents to s-sets
5.4 Notes and references
Exercises
Chapter 6 Projections of fractals
6.1 Projections of arbitrary sets
6.2 Projections of s-sets of integral dimension
6.3 Projections of arbitrary sets of integral dimension
6.4 Notes and references
Exercises
Chapter 7 Products of fractals
7.1 Product formulae
7.2 Notes and references
Exercises
Chapter 8 Intersections of fractals
8.1 Intersection formulae for fractals
8.2 Sets with large intersection
8.3 Notes and references
Exercises
Part II Applications and Examples
Chapter 9 Iterated function systems-self-similar and self-affine sets
9.1 Iterated function systems
9.2 Dimensions of self-similar sets
9.3 Some variations
9.4 Self-affine sets
9.5 Applications to encoding images
9.6 Zeta functions and complex dimensions
9.7 Notes and references
Exercises
Chapter 10 Examples from number theory
10.1 Distribution of digits of numbers
10.2 Continued fractions
10.3 Diophantine approximation
10.4 Notes and references
Exercises
Chapter 11 Graphs of functions
11.1 Dimensions of graphs
11.2 Autocorrelation of fractal functions
11.3 Notes and references
Exercises
Chapter 12 Examples from pure mathematics
12.1 Duality and the Kakeya problem
12.2 Vitushkin's conjecture
12.3 Convex functions
12.4 Fractal groups and rings
12.5 Notes and references
Exercises
Chapter 13 Dynamical systems
13.1 Repellers and iterated function systems
13.2 The logistic map
13.3 Stretching and folding transformations
13.4 The solenoid
13.5 Continuous dynamical systems
13.6 Small divisor theory
13.7 Lyapunov exponents and entropies
13.8 Notes and references
Exercises
Chapter 14 Iteration of complex functions-Julia sets and the Mandelbrot set
14.1 General theory of Julia sets
14.2 Quadratic functions-the Mandelbrot set
14.3 Julia sets of quadratic functions
14.4 Characterisation of quasi-circles by dimension
14.5 Newton's method for solving polynomial equations
14.6 Notes and references
Exercises
Chapter 15 Random fractals
15.1 A random Cantor set
15.2 Fractal percolation
15.3 Notes and references
Exercises
Chapter 16 Brownian motion and Brownian surfaces
16.1 Brownian motion in R
16.2 Brownian motion in Rn
16.3 Fractional Brownian motion
16.4 Fractional Brownian surfaces
16.5 Lévy stable processes
16.6 Notes and references
Exercises
Chapter 17 Multifractal measures
17.1 Coarse multifractal analysis
17.2 Fine multifractal analysis
17.3 Self-similar multifractals
17.4 Notes and references
Exercises
Chapter 18 Physical applications
18.1 Fractal fingering
18.2 Singularities of electrostatic and gravitational potentials
18.3 Fluid dynamics and turbulence
18.4 Fractal antennas
18.5 Fractals in finance
18.6 Notes and references
Exercises
References
Index
