## Spring Semester 2024

### Two examples of renormalization in dynamics, Carlos Matheus, CNRS

Tuesday 9th April at 13:00 - 15:00 in Y35 F47 (UZH)

Thursday 11th April at 13:00 - 15:00 in Y35 F47 (UZH)

Friday 12th April at 10:00 - 12:00 in Y35 F47 (UZH)

**Abstract:** The applications of renormalization ideas in Dynamical Systems became increasingly popular after 1979, and, since then, they played an important role in the study of several classes of low-dimensional systems.

Very roughly speaking, the philosophy of renormalization is that, after appropriate rescalings, the long time behaviors at short scales of certain systems are dictated by other systems within a fixed class S of systems. In particular, such a renormalization procedure can iterated and, as it turns out, the phrase portraits of those systems whose successive renormalizations tend to stay in a compact portion of S can often be reasonably described ("plough in the dynamical plane to harvest in the parameter space", A. Douady).

In this minicourse, we shall illustrate these ideas by explaining the common strategy of "recurrence of renormalization to compact sets" behind two different results:

(a) the solutions of Masur and Veech in 1982 to Keane's conjecture of unique ergodicity of almost all interval exchange transformations;

(b) the solution of Moreira--Yoccoz in 2001 to Palis' conjecture on the prevalence of stable intersections of pairs of dynamical Cantor sets whose Hausdorff dimensions are large.

*His seminar on Wednesday 10.04.2024 is at the usual time and place (13:30-14:30 HG G19.1 (ETH)), for more details see the seminar tab.*

## Autumn Semester 2023

### Lattices, subspaces and diophantine approximation, Nicolas de Saxcé, CNRS and Université Paris-Nord

Monday 18th December at 15:00 - 16:30 in Y27 H25 (UZH)

Tuesday 19th December at 13:00 - 15:00 in Y27 H35/36 (UZH)

Wednesday 20th December at 13:00 - 15:00 in Y27 H46 (UZH)

**Abstract:** Since the work of Minkowski in the early twentieth century, the space of lattices has been a fundamental tool in the study of natural or rational numbers. Then, Margulis and his followers, in particular Dani, showed that methods from ergodic theory could be used very efficiently in that setting. More recently, Schmidt and Summerer started the "parametric geometry of numbers", which is a way to describe diagonal orbits in the space of lattices, using a simple combinatorial coding.
The goal of this mini-course is to introduce the main concepts of parametric geometry of numbers, and to use them to study two problems going back to Jarník and Schmidt:

*Jarník:*Given $r>(n+1)/n$, does there exist $x$ in $\mathbb{R}^n$ such that the inequality $|x-p/q|<q^{-r}$ has infinitely many solutions $p/q$ in $\mathbb{Q}^n$, but for all $c<1$, the inequality $|x-p/q|<cq^{-r}$ has only finitely many solutions $p/q$ in $\mathbb{Q}^n$? (And what is the Hausdorff dimension of the set of such points?)*Schmidt:*Given an $l$-dimensional subspace $x$ in $\mathbb{R}^d$, for what values of $r$ can one always find an $l$-dimensional rational subspace $V$ in $\mathbb{Q}^d$ arbitrarily close to $x$ and such that the distance to x satisfies $d(v,x)<H(v)^{-r}$? (Height and distance on Grassmann varieties will be defined in the first lecture.)

## Autumn Semester 2022

### Exact dimension of Oseledets measures, François Ledrappier, CNRS

Tuesday 22nd November at 15:00 - 17:00 in Y03 G91 (UZH)

Thursday 24nd November at 14:00 - 16:00 in Y03 G91 (UZH)

**Abstract:** The seminar and the mini-course will report on an ongoing joint work with Pablo Lessa (Montevideo). We consider a random walk on a group of matrices. Under suitable assumptions, Oseledets Theorem yields numbers (the Lyapunov exponents) and a random splitting into so-called Oseledets subspaces.
This splitting defines a (random) point in a product of Grassmannians. Our Main result is that the distribution of this point is an exact-dimensional measure. The dimension has a geometric interpretation in terms of the exponents and some partial entropies. The seminar will present the statement and the main partial results. As an example, we also discuss the case when the random walk is supported by an Anosov 3-dimensional representation of a surface group. The lectures will discuss the main details of the proofs.

*This followed a seminar talk on Monday 21st November, with the same title.*

## Spring Semester 2022

### Around Margulis Normal Subgroup Theorem, Uri Bader, Weizmann Institute

Tuesday 24th May at 15:00 - 17:00 in ML F39 (ETH)

Wednesday 25th May at 10:00 - 12:0o in NO C44 (ETH)

**Abstract:**
I will start with presenting a proof of Margulis NST and then discuss various generalizations.

I will relate the above to the dynamics of a group on its space of positive definite functions and the related space of characters.

I will present a variety of new results which are proved using (non-commutative) ergodic theoretical techniques.

## Autumn Semester 2021

### Equidistribution of random walks in tori and applications to super-strong approximation, Weikun He, Korea Institude for Advanced Study (KIAS)

Monday 22nd November at 09:00 - 10:30 in Y21 D68a (UZH)

Wednesday 24th November at 09:00 - 10:30 in Y27 H35/36 (UZH)

Thursday 25th November at 09:00 - 10:30 in Y34 F01 (UZH)

**Abstract:** Given a subgroup of $\operatorname{SL}(d,\mathbb{Z})$, we can look at its reduction modulo arbitrary integers. If we fix a generating set, we obtain a family of corresponding Cayley-Schreier graphs. We can ask whether this family is a family of expander graphs. This property is also known as expansion or spectral gap or super-approximation.

Recently, in a joint work with Nicolas de Saxcé, we proved that the answer is yes if the identity component of the Zariski closure of the group is absolutely simple. The proof is mainly based on the work of Salehi-Golsefidy-Varju, the work of Bourgain-Varju and an improvement of the Bourgain-Furman-Lindenstrauss-Mozes theorem about the equidistribution of random walks in tori.

The goal of this mini-course is to give an introduction to this topic and convey ideas of the main steps of the proof.

*Lecture 1:* Definition of super-approximation and Bourgain-Gamburd expansion machine.

*Lecture 2:* Equidistribution of random walks on the torus and its implication.

*Lecture 3:* Expansion in low levels and how to put things together.