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

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