VO 250138-1 Ergodic Theory I (6 ECTS)

Lecturer: Prof. Henk Bruin and Dr. Davide Ravotti

Email Henk Bruin or Davide Ravotti 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

• discrete time dynamical systems, including circle maps, doubling maps, continued fraction maps.
• Invariant measures, Krylov-Bogul'jubov Theorem.
• Absolute continuity, densities (= Radon-Nikodym derivative)
• Ergodicity and unique ergodicity
• Ergodic Theorems (Birkhoff, von Neumann) and basic applications.
• Normal numbers and Benford's Theorem
• Poincare's Recurrence Theorem, Kac's Lemma.
• Mixing and transfer operators
• shift spaces and Bernoulli meaures
• Perron-Frobenius Theorem
• Cellular automata

References

• Jane Hawkins, Ergodic Dynamics; From Basic Theory to Applications, Springer-Verlag2021, ISBN: 978-3-030-59242-4 (ebook) or ISBN: 978-3-030-59244-8 (paper back)
• 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
Online texts on ergodic theory by:
• Omri Sarig, Lecture Notes on Ergodic Theory Penn State, Fall 2008, in .pdf
• Mike Hochman
• Charles Walkden