VO 250109-1 Ergodic Theory I

Lecturer: Prof. Henk Bruin

Email H. Bruin for further information for this course.

Announcements

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

Assessment

Will be based on an oral exam (in English by default, aber auf Deutsch ist auch möglich ). The material is contained in the class notes: everything up to Section 14: toral autmorphisms, but not the Bernoulli shifts, the Chacon cutting-and-stacking example, and no infinite ergodic theory (Zweimüller's notes).

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.
• 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.
• Zweimüller's notes on infinite ergodic theory (mini-course University of Surrey 2009) in pdf.
• Viana's notes on interval exchange transformations in pdf.