Mathematics of Fractals FRA
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
- 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.