The Ergodic Theory of Dynamical systems is a branch of the theory of Differential Equations. It was founded as a separate discipline by Poincare, and developed by Birkhoff and Smale. Substancial contributions were made by mathematicians of the Moscow school, e.g. Kolmogorov, Arnold, Alekseev, Anosov, Sinai, and others. The proposed course is an introduction to the subject. It contains a survey of the field. The basic theorems are proved; the ideas of the proofs of more difficult results are explained without technical details.

Prerequisites: introductory course to Ordinary Differential Equations.

Curriculum:

1. Dynamical systems: trajectories, simple and strange attractors, chaoticity
2. Action on measures, notion of a transfer operator, invariant measures
3. Ergodicity
4. Birkhoff ergodic theorem
5. Mixing
6. Main ergodic constructions:
   • Direct and skew products
   • Poincare and integral maps
   • Natural extension and noninvertibility
7. Entropy: metric and topological approaches
8. Sinai-Bowen-Ruelle and natural/observable measures
9. Direct operator formalism:
   • Banach spaces of signed measures
   • Ionescu-Tulcea and Marinescu ergodic theorem
   • andom perturbations
10. Spectral theory for Koopman and transfer operators
11. Approximation by finite rank operators
12. Multicomponent systems: phase transitions
13. Mathematical background of numerical simulations

Textbooks

• M. Blank. Ergodic theory of noninvertible transformations, MCCME, Moscow, 2004.
• B. Hasselblat, A. Katok. Introduction to the modern theory of dynamical systems, Cambridge Univ. Press.
• I.P. Kornfeld, S.V. Fomin, Ya.G. Sinai. Ergodic theory, M.:Nauka, 1980. 382 pp. (Springer Verlag, 1982)