Dynamical systems 1000-135UD
Introductory examples, archetypes and definitions. Mappings and flows. Continued fractions and the Gauss map of the interval. Preservation of measure. Conjugacy. Attracting and repelling periodic points. Algebraic diffeomorphisms and flows on the torus and the Weil ergodic theorem. The fundamentals of dynamics on compact spaces. The von Neuman ergodic theorem in Hilbert spaces. Holomorphic maps with a strongly attracting fixed point. One-dimensional dynamics. Homeomorphisms of the circle, their rotation number, Denjoy Theorem. Mappings of the interval and their kneading invariants. Holomorphic maps, the definition of the Julia set and the Mandelbrot set. Ergodic theory. The Birkhoff Ergodic Theorem with applications and generalisations. Ergodic transformations. The Shannon-Breiman-McMillan theorem.The Kryloff-Bogolouboff Theorem. Entropy. Topological and metric entropies as invariants of conjugacy. Smooth hyperbolic systems. Hyperbolic fixed points, invariant manifolds and the Grobman-Hartman theorem. The Poincare return map. Examples of hyperbolic systems, including the Smale horseshoe, the solenoid attractor in 3-space, the Plykin attractor in the plane. General hyperbolic systems and the related invariant foliations. Anosov diffeomorphisms and their basic properties.
Type of course
Course coordinators
Learning outcomes
1. Knowledge of basic notions of Dynamical Systems (dynamical system, trajectory, limit set, conjugacy).
2. Dynamics of circle homeomorphisms: Knowledge of the notion of rotation number and its properties. Knowledge of the Denjoy Theorem.
3. Iteration of interval maps: Knowledge of the Sharkovskii Theorem. Knowledge of basic information on quadratic (logistic) family. Knowledge of the notion of period-doubling bifurcations and the Hyperbolic Density Theorem in the quadratic family.
4. Smooth dynamical systems on manifolds: Knowledge of the Hadamarda-Perron Theorem and definition of stable and unstable manifolds. Knowledge of the Grobmana-Hartman Theorem. Knowledge of the definition of the Morse-Smale systems and their basic properties. Knowledge of the Omega-explosion example. Knowledge of the coding for the Smale horseshoe. Knowledge of the definition of a hyperbolic set and Anosov systems. Knowledge of the Shadowing Theorem. Ability of qualitative analysis of simple examples of smooth dynamical systems.
5. Ergodic theory of dynamical systems: Knowledge of the definition of invariant measure and the notion of ergodicity. Knowledge of basic examples of measure-preserving systems (the Liouville Theorem, geodesic flows, toral automorphisms, billiards). Knowledge of the Birkhoff Ergodic Theorem. Knowledge of the Krylov-Bogoljubov Theorem on the existence of invariant measures. Knowledge of the notion of measure-theoretic entropy and its basic properties. Knowledge of the Shennon-McMillan-Breiman Theorem. Knowledge of the Variational Principle.
6. Holomorphic dynamics: Knowledge of the notion of the Julia set and Mandelbrot set. Knowledge of basic examples of the dynamics of holomorphic maps.
Assessment criteria
Solving homework problems and presenting them during classes. Possible preparing short talks on given subject. Written exam - several problems concerning basic properties and examples of dynamical systems. Oral exam if necessary.
Bibliography
R. L. Devaney: An Introduction to Chaotic Dynamical Systems, Westview Press, Boulder 2003.
S. W. Fomin, I. P. Kornfeld, J. G. Sinaj, Teoria ergodyczna, PWN, Warszawa 1987.
B. Hasselblatt, A. Katok, A First Course in Dynamics. With a panorama of recent developments, Cambridge University Press, New York 2003.
A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambbridge 1995.
V. A. Malyshev, R. L. Minlos, Gibbs random fields. Cluster expansions, Kluwer Academic Publishers Group, Dordrecht 1991 (oryg. rosyjski Nauka, Moskwa 1985).
Z. Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, MIT Press, Cambridge 1971.
C. Robinson, Dynamical systems. Stability, symbolic dynamics and chaos. Studies in Advanced Mathematics, CRS Press, Boca Raton 1999.
W. Szlenk, Gładkie układy dynamiczne, PWN, Warszawa 1982.
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:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: