Basic knowledge of analysis on metric spaces, measure theory and integration is required. We will also need Fourier series and some functional analysis, but it suffices if students learn these topics simultaneously or a bit later.
| week | date | topic |
|---|---|---|
| 1 | 17.02. | Motivation: From initial value problems to dynamical systems, Poincaré's contributions to celestial dynamics, the three body problem, and the birth of modern dynamics; Liouville's theorem; Boltzman's ergodicity hypothesis. First examples: North-South dynamics and the circle rotation. |
| 19.02. | More examples: \(\times p\)-map, one- and two-sided shifts, hyperbolic toral automorphism, continued fraction expansion, "Rationality dedection", Billards, Geodesic flow. Start of Topological Dynamics: Definition of a topological dynamical system, topological transitivity, transitivity of \(\times p\)-map. Chapter 1 in [PY]. | |
| 2 | 24.02. | Equivalent characterizations of topological transitivity, (a special case of the) Baire category theorem, minimality, characterizations of minimality, existence of minimal subsystems, Birkhoff recurrence, minimality of irrational rotation. Chapter 1 in [PY]. |
| 26.02. | Conjugacy of topological dynamical systems. Homeomorphisms of the circle, liftings, orientation preseravation/inversion. Basic properties of liftings of homeomorphisms of the circle, definition of the rotation number as a limsup, proof of convergence of the corresponding sequence. Chapter 6 in [PY]. |
The new exercises will be posted here on Fridays. We expect you to solve the problems in the following week on your own and during the time alloted on Mondays. Exercise sheet \(n\) can be discussed in weeks \(n+1\) and \(n+2\).
| Exercise sheet | Solutions/Hints |
|---|---|
| Exercise sheet 1 | Solutions |
| Exercise sheet 2 |