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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/fractalLeads onto the more advanced sequel Techniques in Fractal Geometry (also by Kenneth Falconer and available from Wiley)

Fractal Geometry: Mathematical Foundations and Applications

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 34*2.3 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 53*3.5 Packing measure and dimensions 54*3.6 Finer definitions of dimension 57*3.7 Dimension prints 58*3.8 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 77*4.4 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 119*8.2 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 139CONTENTS vii9.3 Some variations 1439.4 Self-affine sets 1499.5 Applications to encoding images 155*9.6 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 178*11.2 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 220*13.6 Small divisor theory 225*13.7 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 Rn 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