Math 610D (Ergodic Theory)

Spring 2022

Basic information

  • Instructor: Pablo Shmerkin.

  • Times: MWF 12:00-12:50 (note this was changed from the original time 11:00-11:50).

  • Mode of delivery: Classes are online on zoom at least until January 21st.

  • Office hours: Times will be published on Canvas at the beginning of the semester.

  • Prerequisites: MATH 420/507 or equivalent background in real analysis is essential. Math 404/541 (harmonic analysis) and Math 421/510 (functional analysis) or equivalent would be useful, but the required concepts will be reviewed in the course.

Course description

The course will feature an introduction to Ergodic Theory, which can be viewed as the study of the statistical behaviour (with respect to an underlying measure) of dynamical systems, simply meaning a map T on some space X. In addition to its intrinsic interest, Ergodic Theory has found applications throughout mathematics, notably Combinatorics and Number Theory. There are also close connection to Real and Harmonic Analysis and to Probability Theory.

In the first part of the course we will cover the basic concepts and results in Ergodic Theory. In the last part of the course we will discuss Furstenberg's multiple recurrence theorem and its application to Szemerédi's Theorem. It is unlikely we'll have time to give a complete proof of this theorem, but we will cover several important special cases and discuss the general strategy. Even though this is a result from the 1970s, it started a field of mathematics that is still flourishing today.

Grading policy

I will post exercises throughout the semester (tentatively, 3 sets of exercises). You can choose to either submit at least 75% of the exercises in each set, or give a presentation on a topic chosen together with me near the end of the semester. The presentation will consist of a written paper (5+ pages) and a presentation of the main ideas. It is recommended that you try the exercises even if you choose the presentation. Both the exercises and the presentation can be worked on with other students (and of course questions are welcome) but the final writeup/delivery of the presentation must be done individually.

Course topics (subject to minor revisions)

  1. Measure-preserving systems: definitions and examples.

  2. Recurrence and ergodicity. The mean and pointwise ergodic theorems.

  3. Strong and weak mixing. Spectral characterizations.

  4. Ergodic theory of continuous maps. Ergodic decomposition. Unique ergodicity.

  5. Factors and joinings. Kronecker factors.

  6. Furstenberg's multiple recurrence theorem. Furstenberg's correspondence theorem and Szemerédi's Theorem.

Bibliography

We will roughly follow the book Ergodic Theory (with a view towards number theory) by M. Einsiedler and T. Ward up to Chapter 7 (skipping some topics). This book is available for free online to current UBC students.

There are many other excellent textbooks covering much of the same material, including:

  • Recurrence in Ergodic Theory and Combinatorial Number Theory, by H. Furstenberg.

  • Ergodic Theory, by K. Petersen.

  • An introduction to Ergodic Theory, by P. Walters.

  • Foundations of Ergodic Theory, by M. Viana and K. Oliveira.