Practical information:
We will meet Wednesdays 14-16 in SR 5 at the Mathematikon starting October 19 and ending February 15, 2023.
Students will give 50 minute talks presenting a few pages of the reading or focus on an example. A short problem session (lasting 30 minutes) will conclude each meating.
I will give the first few presentations, and we will assign presentations on the second or third meeting.
Information about credits can be found [muesli] and [offical course page].
Prerequisites:
The seminar assumes knowledge from measure theory and functional analysis (L^p spaces of function classes), linear algebra, and basic topology. A first course in ergodic theory might be helpful, but is not required. Some applications and examples of the theory require a bit of smooth manifold theory, and basic notions in Riemannian geometry (e.g. the geodesic flow).
Description and plan:
Imagine you have a finite collection of 2 x 2 real matrices {A_1, ..., A_n} and a probability measure m assigning to each A_i a number p_i. You would like to know what random matrix products B_k B_{k-1} ... B_1 look like when k tends to infinity and B_j is chosen at random from the list {A_1, ..., A_n} according to the measure m.
Do the matrix entries grow? How fast? Is there a direction in R^2 that the random product of matrices stretches the most? The least? The theory of Lyapunov exponents gives us tools to understand the answers to these questions, almost surely, under suitable hypotheses. Moreover, many concrete problems in dynamics, geometry, topology, and group theory can be modeled on "random matrix products," even deterministic processes such as approximating irrational numbers by their continued fraction expansions.
Although there is a very general theory of Lyapunov exponents for random n x n matrix products, we will focus on the 2 dimensional theory and applications in 1 and 2 dimensional dynamical systems coming from geometry and topology. This will make some of the proofs more manageable, while still seeing many features and interesting applications.
We will review some ergodic theory with emphasis on two a central examples: irrational rotations of the circle and continued fraction expansions. This will help set the stage for the theory of Lyapunov exponents (especially how it relates to continued fraction expansions).
Reading/resources:
Ergodic Theory with a view towards Number Thoery, M. Einsiedler & Thomas Ward (Chapters 2 and 3).
Lectures on Lyapunov Exponents, M. Viana (Excerpts from chapters 1-4)
Notes on the multiplicative ergodic theorem, Simion Filip (https://doi.org/10.1017/etds.2017.68 or https://math.uchicago.edu/~sfilip/public_files/MET_lectures.pdf)
...
Schedule:
Weeks 1 and 2 (James):
Introduction to/review of/definitions and examples of ergodic and measure preserving systems.
Mean and pointwise ergodic theorems
Weeks 3 and 4 (to be assigned):
Elementary properties of continued fractions . Continued fraction map and Gauss measure. (Einsiedler--Ward, Sections 3.1 and Section 3.2 pp. 76-79)
Ergodicicty of the Gauss map and Applications of the ergodic theorems to the continued fraction map. (Einsiedler--Ward, Section 3.2 pp. 79-86)
Weeks 5 & 6 (to be assigned):
Introduction to Lyapunov exponents. Statement of existence theorem for Lyapunov exponents in dimesion 2. Linear cocycles and random matrix multiplication: definition and examples. (Viana, Sections intro to section 1 & sections 1.1, 1.2, intro to section 2 & sections 2.1.1, 2.1.2)
Hyperbolic cocycles: definitions and properties. (Viana, Sections 2.2.1 and 2.2.2)
Weeks 7 and 8(to be assigned):
Functional analysis preliminaries to the proof of Osceledet's Theorem: Theorems of Krylov–Bogoliubov and Banach-Alaoglu.
Proof of Osledet's Thoerem by induction: (Filip, Section 2)
Weeks 9 and 10 (to be assigned):
Applications of Oseledet's theorem and lypunov exponents in dimension 2 (under construction).