Research Seminar "Introduction to Ergodic Theory"
- This course is aimed at providing students with a solid working knowledge in the basic concepts, important techniques and examples in Ergodic Theory of dynamical systems.
- At the end of the course the student is expected be able to analyze statistical properties of dynamical systems, in particular to be familiar with the ergodic theorem and its numerous applications, e.g. in number theory
- Dynamical systems: trajectories, invariant sets, simple and strange attractors and their classification, randomness
- The action in the space of measures, transfer operator, invariant measures. Comparison with Markov chains
- Ergodicity, Birkhoff ergodic theorem, mixing, CLT. Sinai–Bowen–Ruelle measures and natural/observable measure
- Basic ergodic structures: direct and skew products, Poincare and integral maps, a natural extension and the problem of irreversibility
- Ergodic approach to number theoretical problems
- Entropy: metric and topological approaches
- Operator formalism. Spectral theory of dynamical systems. Banach space of measures, random perturbations
- Multicomponent systems: synchronization and phase transitions
- Mathematical foundations of numerical simulations
- Hasselblatt, B., Takens, F., & Broer, H. W. (2010). Handbook of Dynamical Systems. Amsterdam: North Holland. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=344991
- Katok, A. B., & Hasselblatt, B. (2002). Handbook of Dynamical Systems (Vol. 1st ed). Amsterdam: North Holland. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=207259
- Michael Blank. (2018). Discreteness and Continuity in Problems of Chaotic Dynamics. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1790218