Book:Kenneth Falconer/Fractal Geometry: Mathematical Foundations and Applications

Kenneth Falconer: Fractal Geometry: Mathematical Foundations and Applications (3rd Edition)

Published $\text {2014}$, Wiley

ISBN 978-1-119-94239-9

Contents

Preface to the first edition
Preface to the second edition
Preface to the third edition
Course suggestions
Introduction
PART I FOUNDATIONS
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
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
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
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
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
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
7 Products of fractals
7.1 Product formulae
7.2 Notes and references
Exercises
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
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
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
11 Graphs of functions
11.1 Dimensions of graphs
*11.2 Autocorrelation of fractal functions
11.3 Notes and references
Exercises
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
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
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
15 Random fractals
15.1 A random Cantor set
15.2 Fractal percolation
15.3 Notes and references
Exercises
16 Brownian motion and Brownian surfaces
16.1 Brownian motion in $\R$
16.2 Brownian motion in $\R^n$
16.3 Fractional Brownian motion
16.4 Fractional Brownian surfaces
16.5 Lévy stable processes
16.6 Notes and references
Exercises
17 Multifractal measures
17.1 Coarse multifractal analysis
17.2 Fine multifractal analysis
17.3 Self-similar multifractals
17.4 Notes and references
Exercises
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