Falconer K. Fractal Geometry: Mathematical Foundations and Applications

2nd edition. — N. Y.: Wiley, 2003. — 367 p.It is thirteen years since Fractal Geometry - Mathematical Foundations and Applications was first published. In the meantime, the mathematics and applications of fractals have advanced enormously, with an ever-widening interest in the subject at all levels. The book was originally written for those working in mathematics and science who wished to know more about fractal mathematics. Over he past few years, with changing interests and approaches to mathematics teaching, many universities have introduced undergraduate and postgraduate courses on fractal geometry, and a considerable number have been based on parts of this book.
Thus, this new edition has two main aims. First, in indicates some recent developments in the subject, with updated notes and suggestions for further reading. Secondly, more attention is given to the needs of students using the book as a course text, with extra details to help understanding, along with the inclusion of further exercises,
(from Preface to the second edition).Preface.
Preface to the second edition.
Course suggestions.
Introduction.
Notes and references,
Foundations.
Mathematical background.
Basic set theory.
Functions and limits.
Measures and mass distributions.
Notes on probability theory,
Hausdorff measure and dimension.
Hausdorff measure.
Hausdorff dimension.
Calculation of dimension - simple examples.
Equivalent definitions of Hausdorff dimension.
Finer definitions of dimension.
Alternative definitions of dimension.
Box-counting dimensions.
Properties and problems of box-counting dimension.
Modified box-counting dimensions.
Packing measures and dimensions.
Some other definitions of dimension.
Techniques for calculating dimensions.
Basic methods.
Subsets of finite measure.
Potential theoretic methods.
Fourier transform methods.
Local structure of fractals.
Densities.
Structure of 1-sets.
Tangents to s-sets.
Projections of fractals.
Projections of arbitrary sets.
Projections of s-sets of integral dimension.
Projections of arbitrary sets of integral dimension.
Products of fractals.
Product formulae.
Intersections of fractals.
Intersection formulae for fractals.
Sets with large intersection.
Application and examples.
Iterated function systems - self-similar and self-affine sets.
Iterated function systems.
Dimensions of self-similar sets.
Some variations.
Self-affine sets.
Applications to encoding images.
Examples for number theory.
Distribution of digits of numbers.
Continued fractions.
Diophantine approximation.
Graphs of functions.
Dimensions of graphs.
Autocorrelation of fractal functions.
Examples for pure mathematics.
Duality and the Kakeya problem.
Vitushkin's conjecture.
Convex functions.
Groups and rings of fractional dimension.
Dynamical systems.
Repellers and iterated function systems.
The logistic map.
Stretching and folding transformations.
The solenoid.
Continuous dynamical systems.
Small divisor theory.
Liapounov exponents and entropies.
Iteration of complex functions - Julia sets.
General theory of Julia sets.
Quadratic functions - the Mandelbrot set.
Julia sets of quadratic functions.
Characterization of quasi-circles by dimension.
Newton's method for solving polynomial equations.
Random fractals.
A random Cantor set.
Fractal percolation.
Brownian motion and Brownian surfaces.
Brownian motion.
Fractional Brownian motion.
Lévi stable processes.
Fractional Brownian surfaces.
Multifractal measures.
Coarse multifractal analysis.
Fine multifractal analysis.
Self-similar multifractals.
Fractal growth.
Singularities of electrostatic and gravitational potentials.
Fluid dynamics and turbulence.
Fractal antennas.
Fractals in finance.
References.
Index.