*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)*