# Book:Kenneth Falconer/Fractal Geometry: Mathematical Foundations and Applications

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## Kenneth Falconer:

## 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