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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.
Last update: Vlasák Miloslav, RNDr., Ph.D. (10.05.2018)
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The exam is oral. The examination requirements are given by the topics in the syllabus, in the extent to which they they were taught in course. Last update: Kučera Václav, doc. RNDr., Ph.D. (12.05.2018)
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Devaney, R.L.: An introduction to chaotic dynamical systems, Westview Press, 2003. Barnsley, M. F.: Fractals everywhere, Boston: Academic Press Professional, 1993. Beardon, A.F.: Iteration of rational functions, Graduate Texts in Mathematics vol. 132, Springer, 1991. Last update: Kučera Václav, doc. RNDr., Ph.D. (29.10.2019)
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The exam is in oral form. The requirements are given by the scope covered in the lecture. Last update: Kučera Václav, doc. RNDr., Ph.D. (10.06.2019)
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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. Last update: Kučera Václav, doc. RNDr., Ph.D. (12.05.2018)
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Basic general knowledge of mathematical analysis and linear algebra. Last update: Vlasák Miloslav, RNDr., Ph.D. (10.05.2018)
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