Daniel Ingebretson

Growth rates of random vectors

Using the assumptions in Theorem 2.1 of [SW] and the Oseledec theorem, we will prove Propositions 3.1, 3.2, and 3.3 from [SW]. These are statements about growth rates of vectors under random matrix products, as well as limiting directions for these vectors. Here are the notes for the lecture.

Daniel Ingebretson

Growth rates of random vectors

Daniel will complete his talk from last week.

Carlos De La Cruz Mengual

Classification of stationary measures on the torus (the big picture)

Consider a probability measure mu on SL(2,Z) with a finite support that generates a sub-semigroup which acts strongly irreducibly on R^2. Mostly following the Takagi lectures of Benoist and Quint, we will present a proof of the fact that the only atom-free mu-stationary measure on the 2-torus is the Haar probability. In the first part of the talk, we will discuss the statement and assumptions, and explain the big picture of the proof modulo the so-called exponential drift argument. This one will be the focus of the second part.

Notes for Carlos' talk are available here (note, this includes both the notes made during the lecture and the more detailed ones made beforehand).

Notes from Oberwolfach argbeitsgemeinschaft, the relevant part is the notes of a talk by Nicolas de Saxce. But other chapters of the notes are also very useful.

Here is the Zoom recording of Carlos' lecture (will expire in approximately one month).

Weikun He

Leafwise measures

As we saw in the previous talk, the notion of leaf-wise measures is used in the work of Benoist-Quint on the classification of stationary measures. In this talk, I will begin by recalling the definition and properties of conditional measures. Then I will define and construct leaf-wise measures. We will see that leaf-wise measures are generalisations of conditional measures more adapted to group actions.

Here are the links to recordings of Weikun's lectures. First video Second video.

Rene Ruehr

Verifying Conditions (I), (II), (III) and Furstenberg measures

In this talk we shall verify conditions (I), (II) and (III) of Theorem 2.1, [SW] for the random walk governed by a measure mu supported on the upper triangular matrices as in Theorem 1.1, [SW]. To do so we shall recall Oseledets Theorem and what information can be gained by applying it simultaneously to subspaces and quotient spaces.

The elusive space W of Theorem 2.1 will turn out to be the collection of eigenspaces for the positive weights of the diagonal matrices A in the KAU decomposition of the support of mu. Reference: Chapter 6 [SW].

Changing topics, we then discuss the existence and properties of the Furstenberg measures nu_b based on the Martingale Convergence Theorem (which we state without proof). Reference: Lemma 3.2 of [BQ1].

Here are the notes for Rene's talk and links to the Zoom video recording.

Florent Ygouf

Furstenberg measures and stable foliation

In the first part of the talk, I will recall the definition of the Furstenberg measures associated to a stationary measure as well as their principal properties. I will follow the presentation given by Benoist and Quint in sections 3.1 and 3.2 of [BQ1].

I will then prove that the leaves of the stable foliation are given measure 0 by those Furstenberg measure in the context of assumption (I) (II) & (III) of theorem 2.1 of [SW]. This is one of the key preliminary results required to kick off the so-called exponential drift argument that was used by Carlos for the classification of stationary measures on the torus and that will be used again in Manuel's talk. Once again, I will follow the presentation given by Benoist and Quint in section 3.4 of [BQ1].

Here is the link to video of Florent's lecture on Zoom. The password is T1&@9np2

Here are the notes for the lecture.

1. Florent Ygouf

1. continuation of last week's talk: stable leaves have measure zero for the limit measures.

This is the completion of last week's talk. Here are the notes for this lecture.

2. Manuel Luethi

The exponential drift, part I.

Start with a finite set of similarities of the real line with uniform contraction ratio and consider the induced random walk on the unit tangent bundle to the modular surface. We will use the upcoming talks in order to show that the leafwise measures for the Furstenberg measures associated to this random walk are invariant under a fixed unipotent one-parameter subgroup. To this end we will provide a version of the exponential drift argument suitable for this setup.

Here are the typewritten notes for Manuel's lecture, and the notes written during the lecture.

The link to the zoom recording of today's lecture. Password: 1A^k5l%5

Manuel Luethi

The exponential drift, part II.

Manuel will continue with last week's topic.

Here are the typewritten notes for Manuel's lecture (they include all material covered dduring the lecture, and more).

The link to the zoom recording of today's lecture. Password: 4M=*=A!7

Yiftach Dayan

Putting it all together

Yiftach will conclude the proof of the classification of stationary measures, for the special case of the random walk given by Manuel.

Here is a link to recording of the lecture. Password: 4Q+z50qy

Yiftach Dayan and Tsviqa Lakrec

Some remarks on the general case

Yiftach will explain what conclusion one obtains from the exponential drift argument in the general setting of [SW] and [BQ1], and sketch the argument for deriving the classification of stationary measures. Tsviqa will explain the suspension construction which one uses in the exponential drift argument, in a more general case than the one presented in Manuel's talk.

Here is a link to the recording of the lectures. Password is 3u$sVJ92

Tsviqa Lakrec

Remarks on the general case -- suspension construction

Tsviqa will continue his talk from next week, explaining the use of the suspension construction. Here is a link to the recording of the lecture. Password is 5c!!T.D$

Daniel Ingebretson

A special case of quantitative nondivergence

Following Kleinbock-Margulis, we will explain how answering questions about multiplicative diophantine properties of points on manifolds can be reduced to proving quantitative nondivergence results for orbits of unipotent flows on the space of lattices in R^d. We will then prove a special case of quantitative nondivergence for the horocycle flow on lattices in R^2.

Here is a link to the recording. Password: 391469

Niclas Technau

Quantitative nondivergence estimates in higher dimensions

Motivated by Daniel's talk we will prove quantitative nondivergence in arbitrary dimension.

The focus is on the geometric and analytic ideas that underpin the inductive proof of Kleinbock--Margulis.

Here is a link to recording of the lecture . Password: 652967

Erez Nesharim

Kleinbock--Lindenstrauss--Weiss nondivergence of fractals

The phenomena of quantitative nondivergence in the space of unimodular lattices is well known to be robust and useful. In this talk I plan to carefully follow Section 5 in the paper ``On fractal measures and diophantine approximation'' by Kleinbock, Lindenstrauss and Weiss who found a natural class of measures for which it holds. If time allows I will mention a few applications to Diophantine approximation. In particular I will reprove some of the nondivergence results of Kleinbock-Margulis in a more general settings. I will repeat some of the steps in Niclas' lecture, but the lecture will be independent of his.

Link to the KLW paper and to the notes for the talk. Here is a link to recording of the lecture . Password: 343047

Asaf Katz

Using nondivergence estimates to construct bounded trajectories.

This lecture will give an outline of the following paper of Kleinbock and Weiss, in which it is shown that certain fractals contain a full dimension's worth of badly approximable vectors.

Here is a link to recording of the lecture . Password: 522898