Ergodic theory in concerned with the behavior of dynamic systems on the long run and provides statistical statements thereof. It delivers a statistical forecast for systems which are otherwise unpredictlable (e.g., chaotic or genuinely stochastic).
This course discusses the mathematical characterization of this situation. A central role is going to be played by the so-called transfer operator, which describes the action of the dynamics on a distribution of states. We are also going to highlight its importance in applications, when it comes to the numerical and data-driven approximation of quantities of interest.
Here is a preview.
Lecture: Tue 10:15am-11:50am
Exercise: Tue 8:30am - 10am
The lectures will be held online in Webex Meetings. The link to the events will be communicated through the Announcements. Please join with your video turned on and feel free to ask question just like in class with physical presence.
Lecture notes and additional material will be provided in Resources.
Tutorial sessions will be held online via Webex, just like the main lecture. There will be exercise sheets. It is expected that you prepare presentations of solutions to the exercises - you will not be required to submit written solutions. However, for the active participation ("aktive Teilnahme") you will be required to present at least three solutions, at least one in each half of the semester.
Possible ways to participate in the tutorial sessions are sharing your screen and presenting a particular exercise via
- short and concise LaTeXed notes; or
- a readable scan/photo of a handwritten document; or
- deriving a solution live on a note taking app (Microsoft OneNote, Samsung Notes etc.) if you have access to a tablet (preferred); or
- additional ideas are always welcome.
Before presenting in a tutorial session, please perform a screen share in a test session with the Webex application on your platform.
- Calculus (Analysis I-II)
- Linear algebra (Lineare Algebra I-II)
- Basic measure theory (see Handout 1)
- Basic programming (e.g., Matlab or Python)
- Basic concepts from functional analysis are advantageous (duality, L^p spaces, operators)
- [BG] Abraham Boyarsky, Pawel Góra; Laws of Chaos. Springer Science+Business Media New York, 1997
- [BS] Michael Brin and Garrett Stuck; Introduction to Dynamical Systems. Cambridge University Press, 2003
- [LM] Andrzej Lasota and Michael C. Mackey; Chaos, Fractals, and Noise. Springer, 1994
- [Ma] Ricardo Mañé; Ergodic Theory and Differentiable Dynamics. Springer, 1983
- [Sa] Omri Sarig; Lecture Notes on Ergodic Theory
- [Wa] Peter Walters; An Introduction to Ergodic Theory. Springer, 1982