250095 VO Ergodic Theory/Ergodentheorie

Lecturer: Prof. Henk Bruin

Email H. Bruin for further information for this course.

Part of the lectures (March 10 - 26, June 2, June 18-25) will be given by Dr. Dalia Terhesiu (email: D. Terhesiu).

Announcements

On Monday June 2, Dr. Terhesiu will teach the class. On Wednesday June 4, class is canceled.

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, Chacon-Ornstein Theorem and similar results;
- 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
• Jon Aaronson, An Introduction to Ergodic Theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society. ISBN 0-8218-0494-4.
• 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

Assessment

Will be based on an oral exam (in English by default, aber auf Deutsch ist auch möglich ). The oral exam will be about the core material, plus one further topic, listed below (which will be agreed upon when making the appointment for the exam):

Core 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.
Poincare's Recurrence Theorem, Kac's Lemma.

Topic 1:

• Mixing and weak mixing, their characterization and relation.
• Chacon and Kakutani cutting and stacking.

Topic 2:

• Inducing techniques, and properties of induced maps, Folklore Theorem (= expanding circle maps have an invariant density).
• Examples of Manneville-Pomeau, Farey and Gauss map, Boole's transformation, quadratic maps.
• Bernoulli shifts, subshift of finite type, Arnol'd's catmap.

Topic 3:

• Infinite measures and their properties.
• Conservative measures.
• Chacon-Ornstein Theorem, Ratio Ergodic Theorem

Topic 4:

• Transfer operators. Koopman operator. Mean Ergodic Theorem.
• Spectral properties. Toral automorphisms.

Course material (Hand-outs)

• Class notes in pdf. This set of notes may still be updated, and corrected.
• Picture of a Markov Partition.
• Applets and material on Arnol'd's cat map by: Jason Davies, Wikipedia, Saratov group, Math overflow and something else.
• Viana's notes on interval exchange transformations in pdf.