(in Polish) Dynamika holomorficzna

General data

Course ID:	1000-1M14DH
Erasmus code / ISCED:	(unknown) / (unknown)
Course title:	(unknown)
Name in Polish:	Dynamika holomorficzna
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
Course homepage:	http://www.mimuw.edu.pl/~baranski/teach/2018-19/dh.html
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	(unknown)
Type of course:	elective monographs
Short description:	Introduction to the theory of iteration of rational, entire and meromorphic complex functions. Basic notions and methods of holomorphic dynamical systems.
Full description:	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.
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.
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 methods and assessment criteria:	Oral exam or presentation of a talk on a given subject extending the scope of the course

This course is not currently offered.

Course descriptions are protected by copyright.
Copyright by University of Warsaw.