Schedule

Contents of the course

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).

References

• 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

Assessment

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.

Course material (Hand-outs)

• 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.