Ergodic
theory (Fall 2013)
Math 211A (Topics in Analysis)
Course Code: 45025
MWF
Instructor: Anton Gorodetski
Email: asgor@uci.edu
Phone: (949) 824-1381
Office Location: RH 510G
Ergodic theory is the study of statistical properties of dynamical systems relative to a measure on the phase space. Or, in a broader way, it is the study of the qualitative properties of actions of groups on (measure) spaces. Currently ergodic theory is a fast growing field with numerous applications.
This course is an introduction to ergodic theory. Besides classical results on recurrence, convergence of averages, entropy, and mixing properties, we will be also interested in the case when the phase space is equipped with some additional structure (topological or smooth), as well as in various applications (in number theory, mathematical physics, probability). Here is an approximate list of topics to be covered:
- A very short intro/reminder of some elementary measure theory. Measure preserving transformations. Poincare's Recurrence Theorem.
- Ergodicity, basic examples (circle rotations, the doubling map, Bernoulli schemes and Markov chains, cat map). Mixing and weak mixing.
- Birkhoff and von Neumann Ergodic Theorems.
- Metric entropy. The Shannon-McMillan-Breiman Theorem.
- Topological dynamics, existence of invariant measures, variational principle.
- Applications to number theory (the Gauss transformation and Continued Fractions, Szemeredi's Theorem).
- Smooth ergodic theory, Lyapunov exponents, Oseledets Theorem.
The grade will be based on (both take-home) midterm and final exams.
Recommended Texts:
- K.Petersen, Ergodic Theory, any edition.
- P.Walters, An Introduction to Ergodic Theory, any edition.
- A.Katok, B.Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, any edition.
- Ya.G.Sinai, Topics in Ergodic Theory, any edition.
Homework and exams
Links
- Ergodic Theory with a view towards Number Theory, by Manfred Einsiedler and Thomas Ward (free from UCI computers)
- Bryna Kra's lectures on ergodic theory and additive combinatorics
- Curt McMullen's lecture notes on ergodic theory
- Ergodic Theory - Introductory Lectures, by Peter Walters (free from UCI computers)