Syllabus
Fractal geometry: Self-similarity, basic constructions, examples from nature. Hausdorff dimension.
Iterated function systems: Affine self-similar sets, systems of contractions. Existence of the attractor, collage theorem. Algorithms for the generation of attractors, chaos game. Attractor properties.
Iteration of real functions: Bifurcation cascade and diagram. Li-Yorke theorem, Sharkovskii theorem. Quadratic (unimodal) case - definition of chaos, existence of chaotic mappings.
Iteration of complex functions: Quadratic functions, Bernoulli shift, transitivity, sensitivity to initial conditions. Julia and Fatou sets. Examples of the geometry of Julia sets, basic dichotomy. Douady-Hubbard potential, external rays, petals. Mandelbrot set, basic properties, potential, fundamentals of the combinatorics of Mandelbrot's set. Iteration of rational functions, holomorphic dynamics.
Annotation
The course is an introduction to fractal geometry and chaos theory. We will construct the best known types of fractals and derive their basic properties. The key tool will be the concept of iteration. We will focus on iterated function systems (e.g. Barnsley fern), iteration of real functions
(Feigenbaum universality) and iteration of complex functions (Mandelbrot and Julia sets). The course is accessible to a wider range of students of mathematics, as well as physics and computer science.