Dynamical Systems and Ergodic Theory Spring 2024

Lecturer
Manfred Einsiedler
Coordinators
Konstantin Andritsch
Wooyeon Kim
for questions regarding exercise sheets or classes, please contact your tutor
Lectures
Thu 12:15-14:00 in HG E 1.1
Fri 14:15-16:00 in HG E 1.1
Forum
Dynamical Systems and Ergodic Theory

Prerequisites

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.

Lecture Recordings

You can find the recordings of the lecture using the following link.

Lecture Summaries

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

Exercises

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.

Exercise classes

The exercise classes will start in the second week of the semester!
timeroomassistantlanguage
Mo 14-16HG E 21Konstantin Andritsch / Wooyeon Kimen
Tu 10-12HG F 26.5Alexander Furlongen
Tu 12-14HG E 1.1Sauditya Jaiswalen

Literature

Textbooks which can be used as additional reference for some of the topics include: