Ergodic theory (Spring 2018)
Math 296C (Topics in Analysis)
Course Code: 45671
MWF, 12:00pm-12:50pm, RH 340N
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:
1. Invariant
measures, Poincare Recurrence Theorem, ergodicity.
2. Examples:
circle rotations, expanding maps, Arnold cat map, symbolic systems, interval
exchange transformations, etc.
3. Interactions
between topological dynamical systems and ergodic theory, Krylov-Bogolyubov
Theorem, structure of the space of invariant measures, unique ergodicity.
4. Von Neumann
and Birkhoff Ergodic Theorems.
5. Ergodic
Decomposition Theorem.
6. Entropy,
Ornstein Theorem (without proof), Shannon–McMillan–Breiman Theorem.
7. Topological
Entropy and Variational Principle.
8.
Thermodynamical formalism.
9. Oseledets
Ergodic Theorem.
10. Applications
to number theory (e.g. properties of continued fraction expansions),
combinatorics (e.g. van der Warden Theorem), spectral theory
(thermodynamical formalism and Fibonacci Hamiltonian, dynamics of
Schrodinger cocycle and Johnson's Theorem), and probability (e.g. random
matrix products).
The grade will be based on participation and (both take-home) midterm and final exams.
Homework and exams
Midterm (due May 23)
Final (due June 13)
Links
- Ergodic Theory, Hyperbolic Dynamics, and Dimension Theory, by L.Barreira, free to download (from UCI network).
- Foundations of Ergodic Theory, by M.Viana and K.Oliveira, free to download (from UCI network).
- Ergodic Theory with a view towards Number Theory, by M.Einsiendler and T.Ward, free to download (from UCI network).
- Bryna Kra's lectures on ergodic theory and additive combinatorics
- Curt McMullen's lecture notes on ergodic theory