University of Zurich, Rämistrasse 71 (Center campus = Zentrum), at room **KOL-G-217** (G indicates the floor (e.g. floor A, floor B, etc.), 217 is the room number on the G floor).

Note: This is *not* Irchel campus where UZH Mathematics Department is (cocktail will take place at Irchel campus).

**Thursday 8th June **

9:00 - 9:30: Registrations (at KOL-G-217)

9:30 - 10:30: Krikorian

10:30 - 11:00: Coffee break

11:00 - 12:00: Aka

12:00 - 13:30: Lunch break (Mensa)

13:30 - 14:30: Krikorian

14:30 - 15:15: Coffee break

15:15 - 16:15: Crovisier

16:15 - 16:45: Coffee break

16:45 - 17:45: Crovisier

18:15 - : Cocktail (Irchel)

**Friday 9th June**

**9:00** - 10:00: Krikorian (mind **early** timing)

10:00 - 10:30: Coffee break

10:30 - 11:30: Crovisier

11:30 - 11: 50: Coffee break (20 mins)

11:50 - 12:50: Damjanovich

12:50 - 14:30: Lunch break (1h40, Mensa)

14:30 - 15:30: Damjanovich

15:30 - 16:15: Break (**Group picture**, then move to ETH)

16:15 - 17:15: FIM Lecture (Lindenstrauss): Room HG E 5

17:15 - : FIM Cocktail

**Saturday 10th June**

9:00 - 10:00: Aka

10:00 - 10:30: Coffee break

10:30 - 11:30: Aka

11:30 - 11:45: Short break

11:45 - 12:45: Damjanovich (leave the building by 13:00)

**Raphaël Krikorian:***Holomorphic KAM Theorems*

The aim of these lectures is to present in a unified way two topics in KAM theory (Siegel’s linearization Theorem and Moser’s invariant curves theorem) and to show how this can help in understanding a phenomenon discovered numerically by Shigehiro Ushiki on the existence of rotation domains close to a fixed point of a conservative complex Hénon map.

*Lecture 1* — Siegel's Theorem: Let $f:(C^d,0)\to (C^d,0)$ be a holomorphic diffeomorphism admitting the origin as a fixed point. If this fixed point is non resonant in the sense that its eigenvalues are multiplicatively independent, it is not difficult to prove that $f$ is formally linearizable at 0: there exists a formal germ of diffeomorphism conjugating formally $f$ to its linear part $Df(0)$. Poincaré-Dulac theorem gives necessary conditions ensuring that this formal conjugation is indeed convergent. In the case where the eigenvalues of $Df(0)$ are on the unit circle and Diophantine i.e. satisfy a (multiplicative) Diophantine condition the situation is more delicate. However Siegel proved in 1942 by a clever resummation method that the formal linearization map conjugating $f$ to its linear part was holomorphic. I shall present in the first lecture the classical KAM (Kolmogorov-Arnold-Moser) proof of Siegel’s Theorem.

*Lecture 2* — Moser’s Theorem: Consider now a smooth diffeomorphism of the real plane $(R^2,0)$ which is orientation preserving and symplectic (i.e. preserves the area). We assume that the origin is a non resonant elliptic fixed point (its eigenvalues are two different complex conjugate complex numbers on the unit circle which are multiplicatively independent). Moser proved in the 60’ that if in addition $f$ satisfies a twist condition then the origin is accumulated by a positive measure set of invariant curves on which the dynamics of $f$ is conjugated to Diophantine rotations. The proof of this result is based on two ingredients: (a) the Birkhoff Normal Form Theorem which allows to formally linearize $f$ at the origin; (b) the KAM theorem which can be seen as a linearization theorem on a (positive measure) subset of $R^2$. I shall give a proof of this result when $f$ is real analytic and describe how this result can be extended to a more general holomorphic KAM result forholomorphic symplectic diffeomorphisms of $(C^2,0)$.

*Lecture 3* — Some years ago Shigehiro Ushiki (motivated by some questions of E. Bedford) discovered numerically the existence of rotation domains close to a fixed point of a conservative complex Hénon map. These domains are kind of Siegel domains « far away » from the origin. I shall present a theorem which contains in some sense both Siegel’s Theorem and the holomorphic KAM theorem and that explains Ushiki’s phenomenon.

**Danijela Damjanovich:***Introduction to rigidity theory of smooth abelian actions and centralizers of smooth systems*

The rigidity theory mentioned in the title is about the following phenomenon: if a smooth dynamical system has some hyperbolicity, and there is a large group of smooth coordinate changes which do not affect the dynamics, then this may force smooth structure on manifold and may force the action to respect this structure, i.e. to be algebraic. In these three lectures I will talk about algebraic examples, I will describe what we mean by "large group" of coordinate changes for general smooth system and what are useful invariant structures, and will show few methods which lead to rigidity. At the end I will discuss some open problems.

**Sylvain Crovisier:***Non-uniform hyperbolicity of surface diffeomorphisms*

Our aim is to present some results about the ergodic theory of smooth surface diffeomorphisms with positive entropy, which generalize classical properties of uniformly hyperbolic systems. The lectures will focus on a recent tool which controls the variations of the Lyapunov exponents of invariant measures using their entropy. It has been proved in a recent joint work with Jérôme Buzzi and Omri Sarig. The proof is based on reparametrization lemmas which come from Yomdin theory. Among consequences we deduce the exponential mixing of measures maximizing the entropy.

**Menny Aka Ginosar:***Arithmetic and homogeneous dynamics on S-adic spaces*

In this series of talks, I will illustrate how natural arithmetic questions relate to problems in homogeneous dynamics on $p$-adic, $S$-adic, and adelic homogeneous spaces. An accompanying example that will be explored throughout is the sketch of an ergodic-theoretic proof concerning the equidistribution of integer vectors of length $\sqrt{D}$ in $d$-dimensional space, projected onto the unit sphere as $D$ tends to infinity, focusing on the case of $d>3$.

In the first lecture, I will shortly give a dynamics-oriented introduction of the $p$-adic numbers and the adeles. The listener who never learned about $p$-adic numbers is nevertheless *advised to read about their basic properties in some standard text*. We will later describe the geodesic and the so-called "$p$-adic" geodesic flow on the modular surface. To describe the latter and to set the stage for later, we will discuss homogeneous spaces and their $S$-adic extensions. We will then introduce Hecke neighbors and prove their equidistribution using the dynamics of the $p$-adic flow on these $S$-adic extensions.

The second lecture will concentrate on doing the impossible: We will explain how to act with the stabilizer group of an arithmetic object to generate other, closely-related, arithmetic objects. This is a folklore idea, which is quite simple but at first sight might be intimidating. It originates from the study of class groups of algebraic groups and coupled with measure rigidity results it can yield quite powerful results. We will see this technique "in action" by using it to "translate" the above-mentioned equidistribution problem to a statement on measure rigidity on homogeneous spaces.

In the third lecture, we will give a sketch of the needed measure rigidity input: a special case of Gorodnick-Oh's $S$-arithmetic generalization of Mozes-Shah work on the possible weak-* limits measures of a sequence of measures invariant under some unipotents. Depending on time and interest, we might skip the latter and discuss similar arithmetic applications related to “joined” equidistribution problems.