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.

Posted by: matheuscmss | February 26, 2013

Eskin-Kontsevich-Zorich regularity conjecture II: three facts about SL(2,R) and a variant of Rokhlin’s disintegration theorem

As we mentioned in the first post of this series, our goal today is to discuss some elementary facts about ${SL(2,\mathbb{R})}$ and conditional measures. For this sake, we divide this post into two completely independent sections: in the next one we’ll exclusively talk about ${SL(2,\mathbb{R})}$, and in the final section we’ll discuss a variant of the so-called Rokhlin’s disintegration theorem.

Posted by: matheuscmss | February 19, 2013

Eskin-Kontsevich-Zorich regularity conjecture I: introduction

From October 20 to 24, 2011, the conference Dynamics and Geometry organized by H. de Thélin, T.-C. Dinh and C. Dupont took place at Institut Henri Poincaré. This conference was marked by 7 interesting mini-courses by N. Mok, N. Sibony, J.-P. Demailly, A. Zorich, Y. Benoist, Y.-T. Siu and Y. Pesin: the full program is available here.

By the end of the conference, A. Zorich mentioned a conjecture in his groundbreaking article with A. Eskin and M. Kontsevich concerning the regularity of ergodic ${SL(2,\mathbb{R})}$-invariant probability measures on moduli spaces of Abelian differentials.

Right after the end of the conference, Jean-Christophe Yoccoz, Artur Avila and I discussed the possibility of using “soft” approaches to this conjecture in the sense that we wished to stick to “elementary” properties of ${SL(2,\mathbb{R})}$, but not on specific features of translation surfaces.

After a couple of further discussions, mostly in Paris (at Collège de France) and Rio (at IMPA during the first Palis-Balzan conference), we managed to reunite in this preprint here the “soft” elements about ${SL(2,\mathbb{R})}$ and its actions on ${\mathbb{R}^2}$ and moduli spaces leading to a solution of the regularity conjecture of A. Eskin, M. Kontsevich and A. Zorich.

In a series of four posts, we’ll explain the regularity conjecture of Eskin-Kontsevich-Zorich and the “soft” methods in the preprint by A. Avila, J.-C. Yoccoz and myself solving this conjecture.

More precisely, we’ll discuss in today’s post the statement (and motivations) of Eskin-Kontsevich-Zorich’s regularity conjecture and we’ll describe the general lines of our solution of this conjecture. Then, in the next post of the series, we’ll explain the first step in our solution, namely, the proof of 3 elementary facts about ${SL(2,\mathbb{R})}$ and its action on ${\mathbb{R}^2}$ and some classical results about conditional measures. After this, in the third and fourth posts of the series, we’ll give an answer to Eskin-Kontsevich-Zorich’s regularity conjecture by using the results on conditional measures to “transfer” the elementary results about the action of ${SL(2,\mathbb{R})}$ on ${\mathbb{R}^2}$ to the moduli spaces of translation surfaces.

Closing this introduction, let us stress out that, while the Eskin-Kontsevich-Zorich conjecture concerns ${SL(2,\mathbb{R})}$-invariant probabilities in moduli spaces of Abelian differentials, the next post of this series will concern only ${SL(2,\mathbb{R})}$, its action on ${\mathbb{R}^2}$ and conditional measures, and thus it might be of independent interest. In particular, it is not necessary to have prior knowledge of Abelian differentials and translation surfaces to read the second post of this series.

Posted by: matheuscmss | February 17, 2013

Ergodicity of conservative diffeomorphisms (I)

Last Wednesday (February 13, 2013) Sylvain Crovisier gave a talk at the Ergodic Theory seminar at LAGA-Paris 13 (that I’m currently helping to organize) about his joint work with Artur Avila and Amie Wilkinson (still not publicly available yet) on the ergodicity of ${C^1}$-generic conservative (i.e., volume-preserving) diffeomorphisms.

In his talk, Sylvain presented two of the main results of his work with Avila and Wilkinson (see Theorem 2 and Theorem 3 below), and he sketched the proof of one of them (namely, Theorem 2). Then, after his talk, he told me that he plans to discuss the proof of the other main result next Friday (February 22, 2013) at Eliasson-Yoccoz seminar in Jussieu (University Paris 6 and 7).

So, I will proceed as follows: below I’ll reproduce my notes from Sylvain’s talk at LAGA, and, if I manage to take decent notes from Sylvain’s talk at Jussieu, then I’ll complete today’s discussion (in another post) by sketching the proof of the other main result of Avila-Crovisier-Wilkinson (namely, Theorem 3).

As usual, all mistakes/errors in this post are entirely my responsibility.