SUBJECT

Discrete dynamical systems

lecture

master

Faculty of Science

Mathematics MSc

3

Semesters 1-4

Autumn/Spring semester

Topologic transitivity and minimality. Omega limit sets. Symbolic Dynamics. Topologic Bernoulli shift. Maps of the circle. The existence of the rotation number. Invariant measures. Krylov-Bogolubov theorem. Invariant measures and minimal homeomorphisms. Rotations of compact Abelian groups. Uniquely ergodic transformations and minimality. Unimodal maps. Kneading sequence. Eventually periodic symbolic itinerary implies convergence to periodic points. Ordering of the symbolic itineraries. Characterization of the set of the itineraries. Equivalent definitions of the topological entropy. Zig-zag number of interval maps. Markov graphs. Sharkovskii’s theorem. Foundations of the Ergodic theory. Maximal and Birkhoff ergodic theorem.

• A. Katok, B.Hasselblatt: Introduction to the modern theory of dynamical systems.Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press,Cambridge, 1995.
• W. de Melo, S. van Strien,   One-dimensional dynamics, Springer Verlag, New York (1993).
• I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer Verlag, New York, (1981).