Fractals everywhere

Description

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).