TIME AND PLACE:
MWF 10:30-11:20 in SIEG 227
INSTRUCTOR INFORMATION:
Instructor: Boris Solomyak
Office: Padelford C-328, Phone: 685-1307.
E-mail: solomyak "at" math.washington.edu
Office hours: Monday 4-5, Tuesday 11-12 or by appointment.
ELECTRONIC RESOURCES:
- Ergodic theory with a view towards number theory by M. Einsiedler and T. Ward
You can connect to the entire book online (UW restricted) from the Math Library website.
The authors allowed downloading the
Handout
containing the first 4 chapters and appendices.
The bibliography is available
here
- A simple proof of some ergodic theorems by I. Katznelson and B. Weiss, Israel Journal of Mathematics,
Volume 42, Number 4, 291-296, 1982.
Available
here (UW restricted)
- Entropy in Dynamics by M. Einsiedler, E. Lindenstrauss, and T. Ward
This is a book in progress. The authors allowed downloading the
Draft
- Self-similar and self-affine sets and measures by K. Simon and B. Solomyak
This is a book in progress. The chapter on preliminaries is
here
- Dimension theory by C. Bishop and Y. Peres
This is a book in progress. The authors allowed downloading the
Draft
- Notes on Bernoulli convolutions by B. Solomyak
Appeared in Fractal geometry and applications: a jubilee of Benoit Mandelbrot. Part 1, 202-230,
Proc. Sympos. Pure Math., 72, Part 1, Amer. Math. Soc., Providence, RI, 2004. Available
here
RESERVE LIST (overnight loan at the Math. Research Library):
- Ergodic Theory by K. Petersen. Cambridge, 1983.
- An introduction to ergodic theory by P. Walters. Springer, 1982.
- Recurrence in ergodic theory and combinatorial number theory by H. Furstenberg. Princeton, 1981.
- Conformal fractals: ergodic theory methods by F. Przytycki and M. Urbanski. Cambridge, 2010.
- Fractal geometry: mathematical foundations and applications by K. Falconer. Wiley, 2003.
COURSE DESCRIPTION:
Ergodic theory studies measure-preserving transformations, especially the long-term behavior of its orbits.
From its origins in statistical physics and celestial mechanics, it has always been an area at the intersection of many
diverse fields, such as harmonic and functional analysis, probability, and dynamical systems.
It is a very active area with many applications;
one recent example is the celebrated Green-Tao theorem on arithmetic progressions in primes
whose proof uses ideas and tools from ergodic theory. One of the 2010 Fields medalists is Elon Lindenstrauss, who received the Fields
medal "for his results on measure rigidity in ergodic theory, and their
applications to number theory."
The course will be an introduction, with
numerous examples and some applications of number-theoretic nature.
Some results will be presented in detail while others will only be sketched.
The last 3-4 weeks will be devoted to topics in fractal geometry, where ergodic theory
is frequently used as a tool. These topics will include Hausdorff measure and dimension, iterated function systems,
and self-similar sets and measures, including the fascinating family of infinite Bernoulli convolutions.
PREREQUISITES:
Measure theory and integration at the level of graduate Real Analysis. The
background in other fields (such as functional and harmonic analysis, probability and number
theory) is not a prerequisite, but will be introduced as needed.
TENTATIVE PLAN:
The course will follow, though not very closely, the first four chapters of the Einsiedler-Ward book (see the link above), for about half
of the quarter. Other topics will be chosen from various sources listed above. Here is the tentative outline:
- Introduction (history, motivation, connections to other fields)
- Measure-preserving transformations
- Recurrence and Ergodicity
- Von Neumann Mean Ergodic Theorem
- Birkhoff Pointwise Ergodic Theorem
- Strong and weak mixing, spectral properties
- Invariant measure for continuous maps; some topological dynamics
- Applications to Number Theory
- Brief introduction to entropy
- Hausdorff dimension
- Self-similar and self-affine sets, iterated function systems
- Self-similar measures, Bernoulli convolutions
HOMEWORK