Kenneth Falconer

Fractal Geometry

Mathematical Foundations and Applications

• Gebundenes Buch

Produktbeschreibung

This comprehensive and popular textbook makes fractal geometry accessible to final-year undergraduate math or physics majors, while also serving as a reference for research mathematicians or scientists. This up-to-date edition covers introductory multifractal theory, random fractals, and modern applications in finance and science. New research developments are highlighted, such as porosity, while covering other much more sophisticated topics, such as fractal aspects of conformal invariance, complex dimensions, and non-commutative fractal geometry. The book emphasizes dimension in its various forms, but other notions of fractality are also prominent.

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

Produktdetails

• Verlag: Wiley & Sons
• 3. Aufl.
• Seitenzahl: 400
• Erscheinungstermin: 24. Januar 2014
• Abmessung: 236mm x 154mm x 25mm
• Gewicht: 631g
• ISBN-13: 9781119942399
• ISBN-10: 111994239X
• Artikelnr.: 39963563

Autorenporträt

Kenneth Falconer, University of St Andrews, UK.

Inhaltsangabe

Preface to the first edition ix Preface to the second edition xiii Preface
to the third edition xv Course suggestions xvii Introduction xix PART I
FOUNDATIONS 1 1 Mathematical background 3 1.1 Basic set theory 3 1.2
Functions and limits 7 1.3 Measures and mass distributions 11 1.4 Notes on
probability theory 17 1.5 Notes and references 24 Exercises 24 2
Box-counting dimension 27 2.1 Box-counting dimensions 27 2.2 Properties and
problems of box-counting dimension 34 *2.3 Modified box-counting dimensions
38 2.4 Some other definitions of dimension 40 2.5 Notes and references 41
Exercises 42 3 Hausdorff and packing measures and dimensions 44 3.1
Hausdorff measure 44 3.2 Hausdorff dimension 47 3.3 Calculation of
Hausdorff dimension - simple examples 51 3.4 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 60 3.9
Notes and references 63 Exercises 64 4 Techniques for calculating
dimensions 66 4.1 Basic methods 66 4.2 Subsets of finite measure 75 4.3
Potential theoretic methods 77 *4.4 Fourier transform methods 80 4.5 Notes
and references 81 Exercises 81 5 Local structure of fractals 83 5.1
Densities 84 5.2 Structure of 1-sets 87 5.3 Tangents to s-sets 92 5.4 Notes
and references 96 Exercises 96 6 Projections of fractals 98 6.1 Projections
of arbitrary sets 98 6.2 Projections of s-sets of integral dimension 101
6.3 Projections of arbitrary sets of integral dimension 103 6.4 Notes and
references 105 Exercises 106 7 Products of fractals 108 7.1 Product
formulae 108 7.2 Notes and references 116 Exercises 116 8 Intersections of
fractals 118 8.1 Intersection formulae for fractals 119 *8.2 Sets with
large intersection 122 8.3 Notes and references 128 Exercises 128 PART II
APPLICATIONS AND EXAMPLES 131 9 Iterated function systems - self-similar
and self-affine sets 133 9.1 Iterated function systems 133 9.2 Dimensions
of self-similar sets 139 CONTENTS vii 9.3 Some variations 143 9.4
Self-affine sets 149 9.5 Applications to encoding images 155 *9.6 Zeta
functions and complex dimensions 158 9.7 Notes and references 167 Exercises
167 10 Examples from number theory 169 10.1 Distribution of digits of
numbers 169 10.2 Continued fractions 171 10.3 Diophantine approximation 172
10.4 Notes and references 176 Exercises 176 11 Graphs of functions 178 11.1
Dimensions of graphs 178 *11.2 Autocorrelation of fractal functions 188
11.3 Notes and references 192 Exercises 192 12 Examples from pure
mathematics 195 12.1 Duality and the Kakeya problem 195 12.2 Vitushkin's
conjecture 198 12.3 Convex functions 200 12.4 Fractal groups and rings 201
12.5 Notes and references 204 Exercises 204 13 Dynamical systems 206 13.1
Repellers and iterated function systems 208 13.2 The logistic map 209 13.3
Stretching and folding transformations 213 13.4 The solenoid 217 13.5
Continuous dynamical systems 220 *13.6 Small divisor theory 225 *13.7
Lyapunov exponents and entropies 228 13.8 Notes and references 231
Exercises 232 14 Iteration of complex functions - Julia sets and the
Mandelbrot set 235 14.1 General theory of Julia sets 235 14.2 Quadratic
functions - the Mandelbrot set 243 14.3 Julia sets of quadratic functions
248 14.4 Characterisation of quasi-circles by dimension 256 14.5 Newton's
method for solving polynomial equations 258 14.6 Notes and references 262
Exercises 262 15 Random fractals 265 15.1 A random Cantor set 266 15.2
Fractal percolation 272 15.3 Notes and references 277 Exercises 277 16
Brownian motion and Brownian surfaces 279 16.1 Brownian motion in R 279
16.2 Brownian motion in Rn 285 16.3 Fractional Brownian motion 289 16.4
Fractional Brownian surfaces 294 16.5 Lévy stable processes 296 16.6 Notes
and references 299 Exercises 299 17 Multifractal measures 301 17.1 Coarse
multifractal analysis 302 17.2 Fine multifractal analysis 307 17.3
Self-similar multifractals 310 17.4 Notes and references 320 Exercises 320
18 Physical applications 323 18.1 Fractal fingering 325 18.2 Singularities
of electrostatic and gravitational potentials 330 18.3 Fluid dynamics and
turbulence 332 18.4 Fractal antennas 334 18.5 Fractals in finance 336 18.6
Notes and references 340 Exercises 341 References 342 Index 357

Rezensionen

"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)