Organizational meeting

I will the topic for the year and we will discuss the structure of the seminar.

We will discuss basic definitions and results in homogeneous dynamics: constructing a G-invariant metric & measure on a Lie (or lcsc) group G, and, for a subgroup Gamma which is discrete, endowing the quotient G/Gamma with a topological & metrical structure via this construction. Talk will be roughly based on homogeneous dynamics course notes for the course given by Barak in 2013 (warning, the notes may contain inaccuracies). Another reference is this book in preparation by Einsiedler and Ward. Here are the notes from the lecture.

We will discuss Birkhoff's pointwise ergodic theorem, equidistribution, and generic points. We give a brief overview of the needed background in ergodic theory and some examples and applications in number theory. The material of the talk can be found in Einsiedler and Ward, Ergodic theory with a view toward number theory. Here are the notes for the talk.

We will discuss the proof of Raghunathan's measure conjecture by Ratner for the case of SL_2(R). We will introduce Ratner's R-Property and the main ideas of the proof, which we will revisit later in the seminar. The lecture is based on this paper of Marina Ratner.

We will discuss the definition and properties of conditional expectation and conditional measures, as well as prove their existence and uniqueness. The discussion will follow Chapter 5 of Einsiedler - Ward. Here are the notes for the lecture.

We will define discrete-time Martingales, and prove two Martingale convergence theorems, as well as discuss several of their applications in Measure Theory, Ergodic Theory and Probability Theory. The talk will be roughly based on the book "Ergodic theory with a view toward Number Theory", by Einsiedler and Ward. Here are the notes for the lecture.

We will discuss the Oseledec theorem about linear cocycles and Lyapunov exponents. A proof of the theorem for the 2 dimensional case will be provided as well as a discussion of the general case and related results. The talk is based mostly on the book "Lectures on Lyapunov Exponents" by Viana. Here are the notes.

We will discuss a few results of Random Matrix Multiplication, including analysing the scenario (presented by Barak last week) where a vector x is multiplied by iid random matrices in the reversed order, i.e. A_1*A_2*...A_n*x. The talk is based on the textbooks "Products of Random Matrices with Applications to Schrodinger Operators" (Bougerol & Lacroix) and "Random Walks on reductive groups" (Benoist & Quint). Here are the notes for the lecture.

We will define the Leaf-wise measures and see some of their properties. In order to prove their existence, we will introduce a construction which was given by M. Einsiedler and E. Lindenstrauss in this 2023 paper. Prior to the lecture, it is recommended to review the definition of conditional measures and their properties which were shown in a previous lecture. Here are the notes for the talk.

We will discuss the Breiman Law of Large Numbers as well as Stationary Measures and Limit Measures, which we will construct and prove a few results about. The talk will be based on the book of Benoist and Quint, Random walks on reductive groups.

We will prove that the Haar measure is the unique stationary measure for a certain random walk on a torus, generated by affine maps with expanding and diagonal linear parts. Using this we will derive a corollary for digital expansion of typical points in certain fractals. Based on this paper of Dayan, Ganguly and Weiss.