Email H. Bruin for further information for this course.

On December 8 there is no (online ) lecture due to a public holiday (Mariae Immaculate Conception).

The lecture of January 12 and 14 are not taking place. Instead, I encourage you to look at Exercises 23-27 and send them by January 17. I intend to do an exrcise session on January 19 or 21.

Day | Time | Room | from | until | |
---|---|---|---|---|---|

Tuesday | 8:00--9:30 | HS2 | Lecture | 06.10.2020 | 26.01.2021 |

Thursday | 8:00--9:30 | HS2 | Lecture | 01.10.2020 | 28.01.2021 |

This is an introduction to ergodic theory, that is: the study of how invariant measures play a role in dynamical systems.
Topics to be discussed are likely to include

- Invariant measures in various standard examples
(both finite and infinite);

- Ergodicity, unique ergodicity and proving ergodicity;

- Poincaré recurrence and Kac' Lemma;

- Ergodic Theorems;

- Induced transformations, Rokhlin towers and similar results;

- Transfer operators;

- Connections to notions from Probability Theory
(Mixing, Bernoulli processes).

- Peter Walters, An Introduction to Ergodic Theory, Springer-Verlag 1975 ISBN 0-387-95152-0.
- Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete 8. Springer-Verlag, Berlin, 1987. ISBN: 3-540-15278-4
- Daniel Rudolph, Fundamentals of measurable dynamics, Oxford Science Publications, Clarendon Press Oxford 1990 ISBN 0-19-853572-4
- Karl Petersen, Ergodic Theory, Cambridge Studies in Advanced Mathematics, 1983, Cambridge University Press ISBN 0-521-38997-6
- Michael Brin and Garrett Stuck, Introduction to Dynamical Systems, Cambridge University Press 2002, ISBN 0-521-80841-3
- Omri Sarig, Lecture Notes on Ergodic Theory Penn State, Fall 2008, in .pdf

Slides of Lectures

- Lecture 1 on October 1 2020.
- Lecture 2 on October 6 2020.
- Lecture 3 on October 6/8 2020.
- Exercises 1-8 for October 13.
- Lecture 4 on October 15 2020.
- Lecture 5 on October 20 2020.
- Lecture 6 on October 22 2020.
- Exercises 9-14 for October 27.
- Lecture 7 on November 3 2020.
- Lecture 8 on November 5 2020.
- Lecture 9 on November 10 2020.
- Lecture 10 on November 12 2020.
- Lecture 11 on November 17 2020.
- Lecture 12 on November 24 2020.
- Exercises 16-22 for November 26.
- Lecture 13 on December 1 2020.
- Lecture 14 on December 3 2020.
- Lecture 15 on December 10 2020.
- Lecture 16 on December 15 2020.
- Lecture 17 on January 7 2021.
- Lecture 18 on January 19 2021.
- Exercises 23-27 on January 21 2021.

- List of exercises

Will be based on an oral exam (in English by default, aber auf Deutsch ist auch möglich) and sufficient participation in the exercise section (for which I want to reserve one slot every other week).

Material:

- Invariant measures, Krylov-Bogul'jubov Theorem.
- Absolute continuity, densities (= Radon-Nikodym derivative)
- Basic examples of measure preserving transformations (circle rotation, doubling map, full shift)
- Ergodicity and unique ergodicity.
- Birkhoff's Ergodic Theorem and basic applications.
- Transfer operator, Koopman operator, Folklore theorem for interval maps.
- Poincare's Recurrence Theorem, Kac's Lemma.
- Mixing and weak mixing, their characterization and relation.

- Class notes in pdf. This set of notes may still be updated, and corrected.
- Some notes on Information Theory in and Shannon's Source Code Theorem pdf.

Updated January 18 2021