Fractals Everywhere

Part 1 Metric spaces, equivalent spaces, classification of subsets, and the space of fractals: spaces, metric spaces; cauchy sequences; limit points, closed sets, perfect sets and complete metric spaces; compact sets, bounded sets; open sets, interiors and boundaries; connected sets; disconnected sets and pathwise connected sets; the metric space (H(X),h) - the place where fractals live; the completeness of the space of fractals; additional theorems about metric spaces. Part 2 Transformations on metric spaces, contraction mappings and the construction of fractals: transformations on the real line; affine transformations in the Euclidean plane; mobius transformations on the Riemann sphere; analytic transformations, how to change coordinates; the contraction mapping theorem; contraction mappings on the space of fractals, two algorithms for computing fractals from iterated function systems; condensation sets; how to make fractal models with the help of the collage theorem; blowing in the wind - continuous dependence of fractals on parameters. Part 3 Chaotic dynamics on fractals: the addresses of points on fractals; continuous transformations from code space to fractals; introduction to dynamical systems; dynamics on fractals - or how to compute orbits by looking at pictures; equivalent dynamical systems; the shadow of deterministic dynamics; the meaningfulness of inaccurately computed orbits is established by means of a shadowing theorem; chaotic dynamics on fractals. Part 4 Fractal dimension: fractal dimension; the theoretical determination of the fractal dimension; the experimental determination of the fractal dimension; Hausdorff-Besicovitch dimension. Part 5 Fractal interpolation: introduction - applications for fractal functions; fractal interpolation functions; the fractal dimension of fractal interpolation functions; hidden variable fractal interpolation; space - filling curves. Part 6 Julia sets: the escape time algorithm for computing pictures of IFS attractors and Julia sets; iterated function systems whose attractors are Julia sets; the application of Julia set theory to Newton's method; a rich source of fractals - invariant sets of continuous open mappings. Part 7 Parameter spaces and mandelbrot sets: the idea of a parameter space - a map of fractals; Mandelbrot sets for pairs of transformations; the Mandelbrot set for Julia sets; how to make maps of families of fractals using escape times. (Part contents).