Fractal Geometry: Mathematical Foundations and Applications

Description

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) 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