Instructor: Dr. Tamás KELETI

Text: Gerald A. Edgar: Measure, Topology, and Fractal Geometry

Prerequisite: Introductory analysis course with proofs: continuous functions, the limit of a sequence, the sum of an infinite series, etc.; basic abstract set theory: finite vs. infinite sets, countable vs. uncountable sets. Some elementary knowledge of metric spaces and topology is desirable, but a short introduction will be offered to make the course self contained.

Course description: This course provides an introduction into the fractal geometry and its mathematical background: iterated function systems, measures (Lebesgue and Hausdorff measures), dimensions and the topology of metric spaces.

Topics:

• Fractal examples: Cantor set, Sierpinski gasket, Sierpinski carpet, Menger sponge, von Koch snowflake curve, Peano's space-filling curve, dragons, number systems.
• Topology of metric spaces: complete metric spaces, Contraction Mapping theorem, the Hausdorff metric, metric for strings.
• Measure theory: Lebesgue outer measure, Lebesgue measure, genearal measures and outer measures, Hausdorff measure, Hausdorff dimension.
• Self-similar sets: iterated function systems, the existance of self-simiar sets for given similarity maps, similarity dimension and Hausdorff dimension of self-similar sets.
• Fractal sets obtained by iteration of complex functions: Julia sets, the Mandelbrot set. (if time permits)