Instructor: Dr. Zoltán Buczolich
Text: class notes
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).
Course objectives: To understand methods of dealing with discrete dynamical systems. Maesuring chaos using topological entropy.
Prerequisits:Measure and integration theory. Note that abstract integration and measure theory is more than the Theory of Lebesgue integral. For example you need the concept of absolute continuity/singularity of one measure with respect to another, the concept of Radon-Nikodym derivative of an absolutely continuous measure. It is also better if you have seen the Riesz-representation theorem of some positive linear functionals with respect to integration.
It is not a must, but it is better if you have heard of Haar measures on (locally) compact topological groups.
It is also better if you know what does it mean that a set is measurable and you have seen the procedure of (Caratheodory) producing a measure from an outer measure.
You are also expected to know the abstract topological definition of compactness (via coverings).
A short description of the course:
Topological transitivity and minimality. Omega limit sets. Symbolic Dynamics. Topological 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. Lap number of interval maps. Markov graphs. Sharkovskii’s theorem. Foundations of the Ergodic theory. Maximal and Birkhoff ergodic theorem.