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.
You can find the recordings of the lecture using the following link.
| week | date | topic |
|---|---|---|
| 1 | 22.02 | Definition of Dynamical Systems and Examples: times-p-map, hyperbolic toral automorphism, continued fraction expansion, "Rationatlity dedector", Billards, Geodesic flow; Benford's law for powers of 2 |
| 23.02 | Topological Dynamics: topological transitivity, topological mixing, minimality; Baire Category Theorem | |
| 2 | 29.02 | topological mixing, minimal subsystem, van der Waerden Theorem, Multiple Recurrence Theorem |
| 01.03 | proof of Multiple Recurrence Theorem | |
| 3 | 07.03 | proof of van der Waerdens Theorem, Shift map, Furstenbergs Correspondence Principle |
| 08.03 | Symbolic Dynamics, Vertex Shifts, Shifts of finite type | |
| 4 | 14.03 | Morse-Hedlund Theorem, Sturmian Shift, Poincaré Recurrence |
| 15.03 | Koopman Operator, von Neumann Mean Ergodic Theorem, Conditional Expectation Operator | |
| 5 | 21.03 | Conditional Expectation Operator, Birkhoff pointwise ergodic theorem |
| 22.03 | Ergodic transformation/measure, Characterization of Ergodicity, Examples of Ergodic Systems | |
| 6 | 28.03 | Characterization of Ergodicity, (Strong) Mixing and Weak Mixing, Characterization of weak mixing |
| 7 | 11.04 | Characterization of weak mixing, Spectral theorem |
| 12.04 | Advertisement: Continued Fraction Expansions, Existence of invariant measures | |
| 8 | 18.04 | Interplay between Topological Dynamics and Ergodic Theory, Existence and Characterization of Ergodic measures, Ergodic Decomposition |
| 19.04 | Density of Ergodic measures for 2x-map, generic points, Characterization of unique ergodicity, Weyl's Theorem | |
| 9 | 25.04 | Introduction to the Hyperbolic plane |
| 26.04 | Geodesic flow on Hyperbolic plane, Proof of Furstenberg's Theorem | |
| 10 | 02.05 | Geodesic flow, Horocycles |
| 03.05 | Fundamental domain of the modular curve, Measure on the modular curve, Horocycle flow | |
| 11 | 09.05 | Ascension Day -- No lecture |
| 10.05 | aside on Haar measure, Ergodicity of left multiplication, measure on the quotient space X, G-ergodicity on X, Unitary Representation | |
| 12 | 16.05 | Unitary Representation, Ergodicity (I) of geodesic flow, Ergodicity (II) of horocycle flow, Mixing (Howe-Moore) of geodesic flow, Sarnak's Theorem, Unique ergodicity of horocycle flow for uniform lattices (Furstenberg's Theorem), Dani's Theorem |
| 17.05 | Introduction to Entropy Theory, Jensens inequality, maximality of entropy | |
| 13 | 23.05 | Properties of (conditional) Information function and (conditional) Entropy, Dynamical Entropy, Future formula |
| 24.05 | Generating sigma algebras, Entropy is an invariant, Kolmogorov–Sinai Theorem | |
| 14 | 30.05 | Recap of Kolmogorov–Sinai Theorem, Entropy of times-p map, Information on the exam, Shannon-McMillan-Breiman Theorem |
The new exercises will be posted here on Tuesdays. We expect you to solve the problems in the following week and prepare solutions for the next exercise class a week later.
Further you have the possibility to hand in your solutions using the SAMup-Tool. The feedback to your solutions will be uploaded there as well a few days after the deadline.
Information regarding the SAMup-Tool and its usage can be found here: README.
Finally, if you wish to present your solutions during the exercise classes please indicate that using the vorxn-Tool. Presenting solutions is not mandatory but you will definitely profit from explaining your ideas to your colleagues -> Do not miss the opportunity to do so!
Note that you can present in total at most twice one of the first two exercises on the exercise sheets. Other than that there are no restrictions.
| exercise sheet | due by | upload link | your solutions |
|---|---|---|---|
| Exercise sheet 1 | February 27 | submission | overleaf solutions |
| Exercise sheet 2 | March 05 | submission | overleaf solutions |
| Exercise sheet 3 | March 12 | submission | overleaf solutions | Exercise sheet 4 | March 19 | submission | overleaf solutions |
| Exercise sheet 5 | March 26 | submission | overleaf solutions |
| Exercise sheet 6 | April 09 | submission | overleaf solutions |
| Exercise sheet 7 | April 16 | submission | overleaf solutions |
| Exercise sheet 8 | April 23 | submission | overleaf solutions |
| Exercise sheet 9 | April 30 | submission | overleaf solutions |
| Exercise sheet 10 | May 07 | submission | overleaf solutions |
| Exercise sheet 11 | May 14 | submission | overleaf solutions |
| Exercise sheet 12 | May 21 | submission | overleaf solutions |
| Exercise sheet 13 | May 28 | submission | overleaf solutions |
| Exercise sheet 14 | June 04 | submission | overleaf solutions |
20/03/24: Problem 3 in Exercise sheet 5 was replaced with another problem about the conditional expectation. Please check the updated version.
10/04/24: Problem 3 in Exercise sheet 7: the assumption of T being ergodic was missing.
| time | room | assistant | language |
|---|---|---|---|
| Mo 14-16 | HG E 21 | Konstantin Andritsch / Wooyeon Kim | en |
| Tu 10-12 | HG F 26.5 | Alexander Furlong | en |
| Tu 12-14 | HG E 1.1 | Sauditya Jaiswal | en |