Falconer K. Fractal Geometry. Mathematical Foundations and Applications

Foundations 1.
mathematical background.
Basic set theory.
Functions and limits.
Measures and mass distributions.
Notes on probability theory.Exercises.
Hausdorff measure and dimension.
Hausdorff measure.
Hausdorff dimension.
Calculation of Hausdorff dimension — simple examples.
*2.4 Equivalent definitions of Hausdorff dimension.
*2.5 Finer definitions of dimension.Exercises.
Alternative definitions of dimension.
Box-counting dimensions.
Properties and problems of box-counting dimension.
*3.3 Modified box-counting dimensions.
*3.4 Packing measures and dimensions.
Some other definitions of dimension.Exercises.
Techniques for calculating dimensions.
Basic methods.
Subsets of finite measure.
Potential theoretic methods.
*4.4 Fourier transform methods.Exercises.
Local structure of fractals.
Densities.
Structure of 1-sets.
Tangents to s-sets.Exercises.
Projections of fractals.
Projections of arbitrary sets.
Projections of s-sets of integral dimension.
Projections of arbitrary sets of integral dimension.Exercises.
Products of fractals.
Product formulae.Exercises.
ntersections of fractals.
ntersection formulae for fractals.
*8.2 Sets with large intersection.Exercises.
Applications and examples.
terated function systems — self-similar and self-affine sets.
terated function systems.
Dimensions of self-similar sets.
Some variations.
Self-affine sets.
Applications to encoding images.Exercises.
Examples from number theory.
Distribution of digits of numbers.
Continued fractions.
Diophantine approximation.Exercises.
Graphs of functions.
Dimensions of graphs.
*11.2 Autocorrelation of fractal functions.Exercises.
Examples from pure mathematics.
Duality and the Kakeya problem.
tushkin’s conjecture.
Convex functions.
Groups and rings of fractional dimension.Exercises.
Dynamical systems.
Repellers and iterated function systems.
The logistic map.
Stretching and folding transformations.
The solenoid.
Continuous dynamical systems.
*13.6 Small divisor theory.
*13.7 Liapounov exponents and entropies.Exercises.
teration 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.Exercises.
Random fractals.
A random Cantor set.
Fractal percolation.Exercises.
Brownian motion and Brownian surfaces.
Brownian motion.
Fractional Brownian motion.
L ´evy stable processes.
Fractional Brownian surfaces.Exercises.
Multifractal measures.
Coarse multifractal analysis.
Fine multifractal analysis.
Self-similar multifractals.Exercises.
Physical applications.
Fractal growth.
Singularities of electrostatic and gravitational potentials.
Fluid dynamics and turbulence.
Fractal antennas.
Fractals in finance.Exercises.