Fractal Geometry

Kenneth Falconer, University of St Andrews, UK.

• Preface to the first edition ixPreface to the second edition xiiiPreface to the third edition xvCourse suggestions xviiIntroduction xixPart I Foundations 11 Mathematical background 31.1 Basic set theory 31.2 Functions and limits 71.3 Measures and mass distributions 111.4 Notes on probability theory 171.5 Notes and references 24Exercises 242 Box-counting dimension 272.1 Box-counting dimensions 272.2 Properties and problems of box-counting dimension 342.3 Modified box-counting dimensions 382.4 Some other definitions of dimension 402.5 Notes and references 41Exercises 423 Hausdorff and packing measures and dimensions 443.1 Hausdorff measure 443.2 Hausdorff dimension 473.3 Calculation of Hausdorff dimension – simple examples 513.4 Equivalent definitions of Hausdorff dimension 533.5 Packing measure and dimensions 543.6 Finer definitions of dimension 573.7 Dimension prints 583.8 Porosity 603.9 Notes and references 63Exercises 644 Techniques for calculating dimensions 664.1 Basic methods 664.2 Subsets of finite measure 754.3 Potential theoretic methods 774.4 Fourier transform methods 804.5 Notes and references 81Exercises 815 Local structure of fractals 835.1 Densities 845.2 Structure of 1-sets 875.3 Tangents to s-sets 925.4 Notes and references 96Exercises 966 Projections of fractals 986.1 Projections of arbitrary sets 986.2 Projections of s-sets of integral dimension 1016.3 Projections of arbitrary sets of integral dimension 1036.4 Notes and references 105Exercises 1067 Products of fractals 1087.1 Product formulae 1087.2 Notes and references 116Exercises 1168 Intersections of fractals 1188.1 Intersection formulae for fractals 1198.2 Sets with large intersection 1228.3 Notes and references 128Exercises 128Part II Applications and Examples 1319 Iterated function systems – self-similar and self-affine sets 1339.1 Iterated function systems 1339.2 Dimensions of self-similar sets 1399.3 Some variations 1439.4 Self-affine sets 1499.5 Applications to encoding images 1559.6 Zeta functions and complex dimensions 1589.7 Notes and references 167Exercises 16710 Examples from number theory 16910.1 Distribution of digits of numbers 16910.2 Continued fractions 17110.3 Diophantine approximation 17210.4 Notes and references 176Exercises 17611 Graphs of functions 17811.1 Dimensions of graphs 17811.2 Autocorrelation of fractal functions 18811.3 Notes and references 192Exercises 19212 Examples from pure mathematics 19512.1 Duality and the Kakeya problem 19512.2 Vitushkin’s conjecture 19812.3 Convex functions 20012.4 Fractal groups and rings 20112.5 Notes and references 204Exercises 20413 Dynamical systems 20613.1 Repellers and iterated function systems 20813.2 The logistic map 20913.3 Stretching and folding transformations 21313.4 The solenoid 21713.5 Continuous dynamical systems 22013.6 Small divisor theory 22513.7 Lyapunov exponents and entropies 22813.8 Notes and references 231Exercises 23214 Iteration of complex functions – Julia sets and the Mandelbrot set 23514.1 General theory of Julia sets 23514.2 Quadratic functions – the Mandelbrot set 24314.3 Julia sets of quadratic functions 24814.4 Characterisation of quasi-circles by dimension 25614.5 Newton’s method for solving polynomial equations 25814.6 Notes and references 262Exercises 26215 Random fractals 26515.1 A random Cantor set 26615.2 Fractal percolation 27215.3 Notes and references 277Exercises 27716 Brownian motion and Brownian surfaces 27916.1 Brownian motion in R 27916.2 Brownian motion in R n 28516.3 Fractional Brownian motion 28916.4 Fractional Brownian surfaces 29416.5 Lévy stable processes 29616.6 Notes and references 299Exercises 29917 Multifractal measures 30117.1 Coarse multifractal analysis 30217.2 Fine multifractal analysis 30717.3 Self-similar multifractals 31017.4 Notes and references 320Exercises 32018 Physical applications 32318.1 Fractal fingering 32518.2 Singularities of electrostatic and gravitational potentials 33018.3 Fluid dynamics and turbulence 33218.4 Fractal antennas 33418.5 Fractals in finance 33618.6 Notes and references 340Exercises 341References 342Index 357

“Falconer’s book is excellent in many respects and the reviewer strongly recommends it. May every university library own a copy, or three! And if you’re a student reading this, go check it out today!.”  (Mathematical Association of America, 11 June 2014)