Fractal Geometry. Mathematical Foundations and Applications

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