(in Polish) Wstęp do układów dynamicznych

elective courses

The course is an introduction to some aspects of the dynamical systems theory based on the analysis of some model examples. This includes a description of the dynamics of transformations of the interval, circle, torus and complex plane.

1. Dynamics of interval maps based on the quadratic family example – conjugacy, hyperbolicity, symbolic dynamics, Sharkovskii Theorem.

2. Circle homeomorphisms – rotation number, Denjoy Theorem, structural stability, Morse–Smale property.

3. Dynamics of transformation of the torus – translations, algebraic automorphisms, Markov partition.

4. Chaotic systems – Smale horseshoe, examples of attractors, solenoids, stable and unstable manifolds, hyperbolicity.

5. Invariant measures, Poincare Recurrence Theorem, ergodicity, entropy.

6. Examples of billiards – billiards in polygons and ellipse.

7. Holomorphic dynamics – Julia sets, complex quadratic family, Mandelbrot set, complex Newton root-finding method.

8. Hausdorff dimension and fractals.

1. A. Boyarsky and P. Góra, Laws of chaos. Invariant measures and dynamical systems in one dimension, Birkhauser, 1997.

2. R. Devaney, An introduction to chaotic dynamical systems, Westview Press, 2003.

3. B. Hasselblatt, A. Katok, A first course in dynamics. With a panorama of recent developments, Cambridge University Press, 2003.

4. M. Pollicott and M. Yuri, Dynamical systems and ergodic theory, Cambridge University Press, 1998l.

5. C. Robinson, Dynamical systems. Stability, symbolic dynamics and chaos, CRC Press, 1998.

6. W. Szlenk, Wstęp do teorii gładkich układów dynamicznych, Państwowe Wydawnictwo Naukowe, 1982.

1. Knowledge on the basic properties of dynamical system theory (dynamical system, trajectory, limit set, conjugacy).

2. Iteration of interval maps: knowledge of the Sharkovskii Theorem, knowledge of basic information on the quadratic (logistic) family.

3. Dynamics od circle homeomorphisms: knowledge of the notion of rotation number and its properties, knowledge of the Denjoy Theorem.

4. Dynamics of torus transformations: knowledge of basic information on algebraic toral automorphisms.

5. Chaotic dynamical systems: knowledge of the Hadamard-Perron Theorem and the definitions of stable and unstable manifolds, hyperbolic system and attractor, knowledge of the notion of coding for the Smale horseshoe, ability of carrying out qualitative analysis of simple examples of smooth dynamical systems.

6. Ergodic theory of dynamical stystems: knowledge of the definition of invariant measure and the notion of ergodicity, knowledge on the basic examples of measure-preserving dynamical systems, knowledge of the Poincare Recurrence Theorem.

7. Holomorphic dynamics: knowledge of the notion of the Julia and Mandelbrot sets, knowledge on the basic examples of dynamics of holomorphic maps.

Solving homework problems and presenting them during classes. Written exam – several problems concerning basic properties and examples of dynamical systems. Oral exam if necessary.