(in Polish) Dynamika holomorficzna 1000-1M14DH
The course will present basic notions and methods of the theory of iteration of holomorphic functions (polynomials, rational, entire and meromorphic functions) on the complex plane. This theory, started in the 1920-1930 by P. Fatou and G. Julia, has been developing intesively since 1980, in relation with the progress of computer techniques which enable to visualize complicated fractal objects. The plan of the course is as follows.
1. Introduction - examples of the dynamics of holomorphic functions.
2. Local behaviour of a holomorphic function near a fixed point - attracting, repelling, rational and irrational neutral points.
3. Julia sets of holomorphic maps - basic properties.
4. Structure of the Fatou set. Basins of attracting and parabolic orbits, Siegel discs, Herman rings, wandering domains. Classification Theorem. Sullivan's Theorem.
5. Critical points and the dynamics of a map, hyperbolic Julia sets.
6. Quadratic family - the Mandelbrot set, bifurcations.
7. Newton's method of finding zeroes of holomorphic functions.
8. Complex exponential family. Topological and geometric properties of the Julia sets. Dimension paradox.
9. Other questions according to students' interests.
Type of course
Learning outcomes
Knowledge of basic notions and results of the theory of iteration of holomorphic functions (polynomials, rational, entire and meromorphic functions). Knowledge of techniques used in the analysis of the dynamics of such maps. Ability of individual analysis of scientific literature concerning these quastions.
Assessment criteria
Oral exam or presentation of a talk on a given subject extending the scope of the course
Bibliography
A. Beardon, Iteration of Rational Functions, Springer-Verlag, New York, 1991.
W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. Avaliable online as preprint.
L. Carleson, T. Gamelin, Complex dynamics, Springer-Verlag, New York, 1993.
J. Milnor, Dynamics in one complex variable, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, 2006. Available online as preprint.
F. Przytycki, J. Skrzypczak, Wstęp do teorii iteracji funkcji wymiernych na sferze Riemanna, preprint IM PAN 30, 1993. Version without figures.
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: