Posted by: matheuscmss | June 11, 2013

## Finiteness of algebraically primitive closed SL(2,R)-orbits in moduli spaces

Today I gave a talk at the Second Palis-Balzan conference on Dynamical Systems (held at Institut Henri Poincaré, Paris). In fact, I was not supposed to talk in this conference: as I’m serving as a local organizer (together with Sylvain Crovisier), I was planning to give to others the opportunity to speak. However, Jacob Palis insisted that everyone must talk (the local organizers included), and, since he is the main organizer of this conference, I could not refuse his invitation.

Anyhow, my talk concerned a joint work with Alex Wright (currently a PhD student at U. of Chicago under the supervision of Alex Eskin) about the finiteness of algebraically primitive closed ${SL(2,\mathbb{R})}$-orbits on moduli spaces (of Abelian differentials).

Below the fold, I will transcript my lecture notes for this talk.

Posted by: matheuscmss | May 31, 2013

## Dynamical degrees (after Blanc and Cantat)

About one week ago (May 23, 2013), I saw a very nice talk of Serge Cantat on Dynamical degrees at the seminar of the Topology and Dynamics team of Orsay.

Before entering on the details of Serge’s talk, let me mention that, generally speaking, this Topology and Dynamics seminar has a very interesting format: indeed, prior to the “main” talk, some local member makes an informal talk (where we are served with coffee and tea during the exposition…) to make the audience more comfortable with the topic of the main talk. An important feature of this format is that the main speaker is usually not allowed to see the informal talk, so that the main talk usually has some overlaps with the informal talk. Of course, this is a really nice feature since the audience is able to pose more questions after seeing the same idea for the second time (with different perspectives as they were presented by distinct persons).

In the case of Serge’s talk, the informal talk was delivered by Yves de Cornulier who introduced us to several aspects of the so-called Cremona group (the central object in Serge’s talk).

Below the fold, I will reproduce my notes from Serge’s talk around his joint work with Jérémy Blanc, while using from time to time my notes from Yves de Cornulier’s talk. As usual, any mistakes/typos in this post are my entire responsibility.

Posted by: matheuscmss | May 27, 2013

## Cohomology of Schmidt-bounded cocycles

One of the most important results for the measurable study of linear (dynamical) cocycles is Oseledets theorem. Very roughly speaking, this fundamental result says that the fiber dynamics of a linear cocycle ${a:X\rightarrow GL(d,\mathbb{R})}$ (over an invertible map ${T:X\rightarrow X}$ preserving an ergodic probability measure ${\mu}$) is relatively simple in appropriate coordinates (at ${\mu}$-almost every point ${x\in X}$): there exists a decomposition ${\mathbb{R}^d=\oplus_{i=1}^k V_i(x)}$ and a finite collection of numbers ${\lambda_1>\dots>\lambda_k}$ such that ${\|a(T^{n-1}(x))\dots a(x)v_i\|\sim \exp(\lambda_i+o(1))\|v_i\|}$ for every ${v_i\in V_i-\{0\}}$ as ${n\rightarrow\pm\infty}$. The content of Oseledets’ theorem become even clearer when combined with Zimmer’s amenable reduction theorem: by putting these results together, one essentially has that, by selecting a basis of ${\mathbb{R}^d}$ “adapted” to ${\oplus_{i=1}^k V_i(x)}$, the cocycle ${a}$ as a “Jordan normal form”

$\displaystyle a=\left(\begin{array}{ccc} c_1&\ast&\ast \\ 0 & \ddots & \ast \\ 0 & 0 & c_m\end{array}\right) \ \ \ \ \ (1)$

where ${c_n}$, ${n=1,\dots, m}$, are conformal matrices (or rather cocycles), i.e., scalar multiples of orthogonal matrices.

For several applications, this Jordan normal form is satisfying, but sometimes it is desirable to “improve” this normal form in certain specific situations.

For example, the recent paper of A. Eskin and M. Mirzakhani shows a Ratner-like result for ${SL(2,\mathbb{R})}$-actions on moduli spaces of Abelian differentials based on a certain exponential drift argument of Y. Benoist and J.-F. Quint (among several other arguments).

As it turns out, the exponential drift argument strongly relies on the fact that it is possible to perform time-changes for the Kontsevich-Zorich cocycle (over the ${SL(2,\mathbb{R})}$-dynamics on moduli spaces) trying to make it look like a conformal by blocks linear cocycle. Here, the existence of such time-changes is not a consequence of Oseledets theorem and Zimmer’s amenable reduction theorem: indeed, it is possible to check that the existence of these time-changes are essentially equivalent to have that all ${\ast}$ blocks in (1) vanish.

In particular, we see that it is important (for some profound applications) to dispose of a criterion ensuring that a linear (dynamical) cocycle is conformal by blocks possibly after an adequate (measurable) change of coordinates.

In this post, we will follow this article of K. Schmidt (from 1981) to give a complete answer to the question in the previous paragraph.

Posted by: matheuscmss | May 10, 2013

## A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

Martin Möller, Jean-Christophe Yoccoz and I have just upload to ArXiv our paper “A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces“. In some sense, the main mathematical content of this paper was already discussed in a certain amount of details in this blog (see these five posts here), and, thus, this short post will just give some quick “historical comments” on this paper.

Posted by: matheuscmss | April 29, 2013

## IPAM’s program “Interactions between Analysis and Geometry” and John Pardon’s talk on Hilbert-Smith conjecture for 3-manifolds

Two weeks ago, I was in Los Angeles to attend Workshop II: Dynamics of Groups and Rational Maps of the IPAM program Interactions between Analysis and Geometry.

The workshop was very interesting in several aspects. First, the topics of the talks concerned different research specialities (as you can see from the schedule here), so that it was an excellent opportunity to learn about advances in other related areas. Secondly, the schedule gave sufficient free time so that we could talk to each other. Also, I was happy to meet new people that I knew previously only through their work (e.g., Alex Kontorovich and John Pardon).

In particular, we had two free afternoons on Wednesday and Friday, and I certainly enjoyed both of them: on Wednesday Alex Eskin drove me to the beach and we spent a significant part of the afternoon talking to each other there, and on Friday I went to Getty Center with Sasha Bufetov, Ursula Hamenstadt, Pat Hooper, John Pardon, Federico Rodriguez-Hertz, John Smillie, and Anton Zorich, where, besides classical painters like Monet, Renoir, etc., I saw

As usual, the talks were very nice (and they will be available at IPAM website here in a near future), and hence I decided to transcript in this post my notes of one of the talks, namely, John Pardon’s talk on his solution of Hilbert-Smith conjecture for 3-manifolds. Of course, the eventual mistakes in what follows are my entire responsibility.

Posted by: matheuscmss | April 20, 2013

## Second Palis-Balzan International Symposium on Dynamical Systems & Workshop on Combinatorics, Number Theory and Dynamical Systems

It is a pleasure to announce that the websites of the following two conferences (that I’m helping to organize) are now open to online registration.

The Second Palis-Balzan International Symposium on Dynamical Systems is a part of Project Palis-Balzan – Dynamical Systems, Chaotic Behaviour-Uncertainty, sponsored by the Balzan Foundation, from the prestigious award conferred to Jacob Palis (and IMPA) by the Balzan Foundation in 2010 (with previous winners [in Mathematics category] including A. Kolmogorov, E. Bombieri, J.-P. Serre, A. Borel, M. Gromov and P. Deligne). This project is mainly coordinated by Jacob Palis and Jean-Christophe Yoccoz, and the organizing committee of the Second Palis-Balzan symposium consists of S. Crovisier, J. Palis, Jean-Christophe Yoccoz and myself.

We strongly recommend all potential participants of the Second Palis-Balzan symposium (and especially the ones in the Paris area) to register in the corresponding website: indeed, the staff of Institut Henri Poincaré informed that all conferences will be held in amphithéâtre Hermite whose maximum capacity is 150 persons; thus, it is important for us to have a vague idea of the total number of participants.

The Workshop on Combinatorics, Number Theory and Dynamical Systems is part of a thematic semester on Dynamical Systems to be held at IMPA from August to November 2013. The main organizers of this thematic semester are C. G. Moreira, E. Pujals and M. Viana, and they decided that each month of this semester will be dedicated to a specific topic in Dynamics. In particular, the month of August 2013 will focus on interactions between Combinatorics, Number Theory and Dynamics, and, after some mini-courses, we will have the workshop (organized by C. Mauduit, C. G. Moreira, Y. Lima, J.-C. Yoccoz and myself) mentioned above. Among the confirmed speakers, we have:

• Pierre Arnoux
• Tim Austin
• Vitaly Bergelson
• Julien Cassaigne
• Alex Eskin
• Sébastien Ferenczi
• Albert Fisher
• Bryna Kra
• Yoshiharu Kohayakawa
• Ali Messaoudi
• János Pintz
• Miguel Walsh
• Barak Weiss
• Maté Wierdl
• Luca Zamboni

and we expect to confirm the participation of the following mathematicians:

• Jean Bourgain
• Yann Bugeaud
• Hillel Furstenberg
• Elon Lindenstrauss
• Curtis T. McMullen
• Peter Sarnak

I think that this is all I have to say about these conferences for now (but you can look at their respective webpages for updated information). See you! (in Paris or Rio)

Posted by: matheuscmss | March 26, 2013

## Second Bourbaki seminar of 2013

Last Saturday (March 23, 2013), the second Bourbaki seminar of this year took place at amphithéâtre Hermite of Institut Henri Poincaré (as usual), and the following topics were discussed:

Once more the speakers did a great job in explaining these topics to an audience of non-experts, and, for this reason, I decided to make a post about one of these talk.

Contrary to last time, it was “easy” for me to choose which topic to pick: given my tastes, I had to choose between Ore’s conjecture and ending laminations, and I opted for Ore’s conjecture because the website Images de Mathématiques made available an excellent article (in French) by F. Guéritaud with a guided tour (with plenty of beautiful pictures!) around the works of Masur-Misnky and Brock-Canary-Minsky.

So, below I will transcript my notes of G. Malle’s talk about Ore’s conjecture. As usual, the eventual mistakes in what follows are my entire responsibility.

Posted by: matheuscmss | March 18, 2013

## Ergodicity of conservative diffeomorphisms (II)

Sylvain Crovisier gave on February 22, 2013, a second talk — this time at Eliasson-Yoccoz seminar in Jussieu — about his joint work with Artur Avila and Amie Wilkinson that we started to discuss a few weeks ago. In fact, last time we saw that two of the main results of Avila-Crovisier-Wilkinson are:

Theorem 1 (A. Avila, S. Crovisier and A. Wilkinson) There exists ${\mathcal{G}\subset\textrm{Diff}^{\,1}_v(M)}$ a residual (i.e., ${G_{\delta}}$-dense) subset such that for any ${f\in\mathcal{G}}$:

• (ZE) either all Lyapunov exponents ${\lambda_i(x)}$ of ${v}$-a.e. ${x\in M}$ vanish,
• (NUA) or ${f}$ is non-uniformly Anosov in the sense that
• ${f}$ has a (global) dominated splitting, i.e., there is a decomposition ${TM=E\oplus F}$ into ${Df}$-invariant subbundles such that ${F}$ dominates ${E}$, that is, there exists ${N\geq 1}$ with ${\|Df^N(u)\|\leq (1/2)\|Df^N(v)\|}$ for any ${u\in E}$, ${v\in F}$ unitary vectors (“the largest expansion along ${E}$ is dominated by the weakest contraction in ${F}$, but, a priori, neither ${E}$ is assumed to be contracted nor ${F}$ is assumed to be expanded”).
• for ${v}$-a.e. ${x\in M}$, the fibers of ${E_x}$ and ${F_x}$ of the dominated splitting coincide with the stable and unstable Oseledets subspaces, i.e., ${E_x=\mathcal{E}_x^s}$ and ${F_x=\mathcal{E}_x^u}$,

and ${v}$ is ergodic.

Theorem 2 (A. Avila, S. Crovisier, A. Wilkinson) For ${r>1}$, the set of ergodic diffeomorphisms in ${\textrm{Diff}^{\,r}_v(M)}$ contains a ${C^1}$-open, ${C^1}$-dense subset of the set ${PH^r_v(M)}$ of partially hyperbolic volume-preserving ${C^r}$-diffeomorphisms.

Furthermore, we saw a sketch of proof of Theorem 1 based on Sylvain’s talk at LAGA.

Today we’ll focus exclusively on the proof of Theorem 2 based on Sylvain’s talk at Jussieu (assuming, of course, that the reader is familiar with our previous post on this subject).

Posted by: matheuscmss | March 12, 2013

## Eskin-Kontsevich-Zorich regularity conjecture IV: a perfect cancellation result and end of proof of EKZ conjecture

Today we will complete the description of the solution of Eskin-Kontsevich-Zorich regularity conjecture, that is, we will prove that, for any ${SL(2,\mathbb{R})}$-invariant probability measure ${m}$ on a connected component of a stratum of the moduli space of translation surfaces of genus ${g\geq 2}$, the ${m}$-measure of the set ${\mathcal{C}_2(\rho)}$ of ${M\in\mathcal{C}}$ with two non-parallel saddle-connections of lengths ${\leq\rho}$ is

$\displaystyle m(\mathcal{C}_2(\rho))=o(\rho^2)$

For this sake, let us recall that, in the previous post of this series, we considered an arbitrarily fixed level ${X=\{M\in\mathcal{C}: \textrm{sys}(M)=\rho\}}$ of the systole function and we introduced a subset ${X_0^*\subset X}$ consisting of ${M\in X}$ such that all its non-vertical saddle-connections have length ${>\rho}$. Then, we defined the set

$\displaystyle Y^*=\{g_tR_{\theta}M_0: M_0\in X_0^*, |\theta|<\pi/4, 0

and we studied the ${m}$-measure of the subsets

$\displaystyle Y^*(T):=Y^*\cap\{M\in\mathcal{C}:\textrm{sys}(M)<\rho\exp(-T)\}$

of translation surfaces with systole ${<\rho\exp(-T)}$ that are “accessible” by ${g_t}$ and ${R_{\theta}}$ movements from ${X_0^*}$. In this setting, the results of the previous post of this series can be summarized as follows:

• ${m|_{Y^*}=\cos2\theta dt\,d\theta\,m_0}$, where ${m_0}$ is a finite measure on ${X_0^*}$;
• ${m(Y^*(T))=(1/2)\pi\,m_0(X_0^*)\exp(-2T)}$ for all ${T>0}$;
• ${m_0}$ is a density measure in the sense that

$\displaystyle \pi\,m_0(X_0^*)=\lim\limits_{\tau\rightarrow0}\frac{1}{\tau}m(\{M\in\mathcal{C}:\rho\geq\textrm{sys}(M)\geq\rho\exp(-\tau)\})$

From this point, we will divide this final post into two sections. In the first one, we will formalize the idea that ${Y^*(T)}$ occupies most of ${\{M\in\mathcal{C}:\textrm{sys}(M)\leq\rho\exp(-T)\}}$ for adequate choices of ${\rho}$, so that the proof of Eskin-Kontsevich-Zorich regularity conjecture will be reduced to the computation of the ${m}$-measure of ${\mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)}$. Then, in the final section, we will show that the set of ${M_0\in X_0^*}$ such that ${g_t R_{\theta} M_0\in\mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)}$ has small ${m}$-measure for ${T>0}$ large because these ${M_0}$‘s have a pair of non-parallel saddle-connections with a small angle (and ${m|_{Y^*}}$ is ${\cos2\theta dt\, d\theta\,m_0}$).

Posted by: matheuscmss | March 5, 2013

## Eskin-Kontsevich-Zorich regularity conjecture III: accessing deep levels of the systole function

The plan of today’s discussion is to use the tools (“orbit by orbit estimates” and a variant of Rokhlin disintegration theorem) from the previous post to study the following question (stated after the picture).

Let ${X=\{M\in \mathcal{C}: \textrm{sys}(M)=\rho\}}$ be a certain (fixed) level of the systole function on a connected component ${\mathcal{C}}$ of a stratum of the moduli space of translation surfaces of genus ${g\geq2}$. Denote by ${X_0^*}$ the set of ${M\in X}$ such that all non-vertical saddle-connections have length ${>\rho}$ and, for each ${T>0}$, consider the set ${Y^*(T)}$ of translation surfaces ${M}$ with systole ${\textrm{sys}(M)<\rho\exp(-T)}$ having the form ${M=g_t R_{\theta} M_0}$ for some ${M_0\in X}$, ${|\theta|<\pi/4}$ and ${0. In other words, using the notation ${J(T,\theta):=\{t\in\mathbb{R}:\|g_t R_{\theta} e_2\|<\exp(-T)\}}$ introduced in the previous post,

$\displaystyle Y^*(T):=\{M=g_t R_{\theta} M_0: M_0\in X_0^*, |\theta|<\pi/4, t\in J(T,\theta)\}$

Geometrically, ${Y^*(T)}$ consists of the pieces of arcs of hyperbola below the threshold ${\rho\exp(-T)}$ in the figure below:

In this notation, given a ${SL(2,\mathbb{R})}$-invariant probability measure ${m}$ on ${\mathcal{C}}$, we want to compute the ${m-}$measure of ${Y^*(T)}$ in terms of ${X_0^*}$, that is, we want to determine how “fat” is the set ${Y^*(T)}$ of translation surfaces that are “accessible” (via ${g_t}$ and ${R_{\theta}}$ movements) from ${X_0^*}$.

In fact, a precise answer to this question will occupy this entire post and, in the next (and last) post of this series, we will use this answer to estimate the ${m}$-measure of the set ${\mathcal{C}_2(\rho\exp(-T))}$ (of translation surfaces with two non-parallel saddle-connections of lengths ${\leq\rho\exp(-T)}$) as follows. Firstly, we will see that ${Y^*(T)}$ captures “almost all” translation surfaces in ${\{M\in\mathcal{C}: \textrm{sys}(M)\leq\rho\exp(-T)\}}$ (for ${\rho}$, i.e., ${X_0^*}$, conveniently chosen). In particular, we will reduce the problem of measuring ${\mathcal{C}_2(\rho\exp(-T))}$ to the task of estimating ${\mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)}$. Then, we will see that the translation surfaces ${M_0\in X_0^*}$ generating a translation surface ${M=g_t R_{\theta} M_0\in \mathcal{C}_2(\rho\exp(-T))\cap Y^*(T)}$ have a pair of saddle-connections with a very small angle ${\leq\theta_0}$ (with ${\theta_0}$ tending to zero as ${T\rightarrow\infty}$) and it is not hard to see that this angle condition corresponds to a subset of ${X_0^*}$ with arbitrarily small “density” in ${X_0^*}$.

We organize this post into two sections. In the first section we will explain how the variant of Rokhlin disintegration theorem from the previous post of this series can be used to describe the ${m}$-measure of ${Y^*(T)}$ in terms of the measure of ${X_0^*}$ with respect to a certain ${m_0}$ obtained after disintegrating ${m}$ along certain pieces of ${SL(2,\mathbb{R})}$-orbits. Then, in the second section, we will explain why the measure ${m_0}$ obtained from this disintegration process is a sort of “density” measure.