Selected Topics in Dynamical Systems, PhD course in mathematics, spring 2017

Abstract: We will discuss some important tools and techniques in the area of dynamical systems: ergodic theorems, Lyapunov exponents, and entropy (topological and measure theoretic). We will study examples in particular from dynamical systems on (a) homogeneous spaces (i.e. spaces of the form Γ\G where G is a Lie group and Γ is a discrete subgroup), (b) the space of translation surfaces (Teichmüller dynamics), and (c) the space of locally finite point sets in a Euclidean space. These three settings are all the subject of current very active research. The course should be of interest for PhD students also outside the area of dynamical systems; for example there are close connections with topics in probability theory and geometry. I aim to make the course accessible with a minimum of prerequisites other than measure theory.


Main course literature:
Omri Sarig, Lecture Notes on Ergodic Theory.
Dave Witte Morris, Ratner's theorems on unipotent flows, University of Chicago Press, 2005.
Marcelo Viana, Ergodic Theory of Interval Exchange Maps, Rev. Mat. Complut. 19 (2006), 7-100.
Marcelo Viana, Lyapunov Exponents of Teichmüller flows, in Partially hyperbolic dynamics, laminations, and Teichmüller flow, Amer. Math. Soc., Providence, RI, 2007.

We will also work directly from some original research articles:
D. Ruelle, Bol. Soc. Brasil. Mat. 9 (1978), 83-87.
R. Mañé, Ergodic Theory Dynamical Systems 1 (1981), 95-102 (and errata).

Lectures: here.

Preliminary plan of lectures:
1. Introduction
2. Ergodic theorems
3. - // -
4. Ergodic decomposition
5. Introduction to homogeneous dynamics
6. The Subadditive Ergodic Theorem
7. The multiplicative ergodic theorem; Lyapunov exponents
8. - // -
9. Entropy
10. - // -
11. Pesin's entropy formula (following the papers by Ruelle and Mañé)
12. - // -
13. Interval Exchange Transformations; Rauzy-Veech renormalization; Teichmüller flow.
14. - // -
15. - // -
16. Translation surfaces.
17. - // -
18. Lyapunov exponents of Teichmüller flows.

Examination: See here for a list of homework exercises. Each participant taking the course for credit can choose the problems that interest him or her the most and hand in solutions to these. The condition for passing the course is to hand in acceptable solutions for problems for a total of at least "100 pt".
DEADLINE: On May 16 or earlier, please show me evidence that you have seriously started on problems with a total value of at least 80 pt.

Andreas Strömbergsson, Tel. (018) 4713221, e-mail: