Book:Kenneth Falconer/Fractal Geometry: Mathematical Foundations and Applications
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Kenneth Falconer: Fractal Geometry: Mathematical Foundations and Applications (3rd Edition)
Published $\text {2014}$, Wiley
- ISBN 978-1-119-94239-9
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