Posted by: matheuscmss | August 6, 2013

## Hodge-Teichmüller planes and finiteness results for Teichmüller curves

Alex Wright and I have just upload to ArXiv our paper Hodge-Teichmüller planes and finiteness results for Teichmüller curves.

As the title of the paper indicates, this article concerns finiteness results for certain classes of Teichmüller curves (i.e., closed ${SL(2,\mathbb{R})}$-orbits in the moduli spaces of Abelian differentials). For example, one of the results in this paper is the finiteness of algebraically primitive Teichmüller curves in minimal strata of prime genus $\geq 3$.

In fact, some parts of this paper were previously discussed in this blog here, and, for this reason, we will not make further comments on the contents of this article today.

Instead, let us take the opportunity to briefly discuss a “deleted scene” of this paper.

More concretely, at some stage of the paper, Alex and I need to know that there are Abelian differentials/translation surfaces with “rich monodromy” (whatever this means) in each connected component of every stratum of genus ${g\geq 3}$.

In this direction, we ensure the existence of such translation surfaces by an inductive argument going as follows.

Starting with an hypothetical connected component ${\mathcal{H}}$ of genus ${g\geq 3}$ containing only translation surfaces with “poor monodromy” (${g-1}$ orthogonal Hodge-Teichmüller planes in the notation of the paper), we follow and adpat some arguments of Kontsevich-Zorich paper on the classification of connected components of strata (cf. Section 5 of our paper) and we work a little bit with the Deligne-Mumford compactification of moduli spaces (cf. Section 6 of our paper) to show that the “poorness of monodromy” property passes down to all translations surfaces in both connected components ${\mathcal{H}(4)^{hyp}}$ and ${\mathcal{H}(4)^{odd}}$. Here, very roughly speaking, the basic idea is that, if we “degenerate” (e.g., pinch off a curve) in a careful way the translation surfaces in ${\mathcal{H}}$, we can decrease the genus without loosing the “poorness of monodromy” property, so that, if we keep degenerating, then we will find ourselves with plenty of genus ${3}$ translation surfaces with poor monodromy.

However, it is easy to see that the property that all translation surfaces in ${\mathcal{H}(4)^{hyp}}$ and ${\mathcal{H}(4)^{odd}}$ have “poor monodromy” is false: indeed, in Section 4 of our paper, we exhibit explicitely two translation surfaces ${M_{\ast}\in \mathcal{H}(4)^{odd}}$ and ${M_{\ast\ast}\in\mathcal{H}(4)^{hyp}}$ with “rich monodromy”. Thus, the hypothetical connected component ${\mathcal{H}}$ can not exist.

As it turns out, before thinking about this argument based on Kontsevich-Zorich classification of connected components and the features of Deligne-Mumford compactification, Alex and I thought of using the following simple-minded argument. One can check that the “richness of monodromy” property passes through finite branched coverings. Hence, it suffices to produce for each connected component ${\mathcal{H}}$ an explicit finite branched cover of one of the translation surfaces ${M_{\ast}}$ or ${M_{\ast\ast}}$ lying in ${\mathcal{H}}$ to deduce that all connected components of all strata possess translation surfaces with “rich monodromy”.

Remark 1 The strategy in the previous paragraph is very natural: for instance, after reading a preliminary version of our paper, Giovanni Forni asked us if we could not make the arguments in Sections 5 and 6 of our paper (related to Kontsevich-Zorich classification of connected components of strata and Deligne-Mumford compactification) more elementary by taking finite branched covers to produce explicit translation surfaces with rich monodromy on each connected component of every stratum. As we will see below, this does not quite work to fully recover the statements in Section 5 of our paper, but it allows to obtain at least part of our statements.

In particular, around November/December 2012, Alex and I started looking at finite branched covers of ${M_{\ast}}$ and ${M_{\ast\ast}}$ “hitting” all connected components of all strata. Unfortunately, this strategy does not work as well as one could imagine: firstly, there are restrictions on the genera of strata we can reach using finite branched covers (thanks to Riemann-Hurwitz formula), and, secondly, it is not so easy to figure out in what connected component our finite branched cover lives.

Nevertheless, this elementary strategy permits to deduce the following proposition (allowing to deduce partial versions of the statements in Section 5 of our paper). Let ${M_{\ast}\in \mathcal{H}(4)^{odd}}$ and ${M_{\ast\ast}\in\mathcal{H}(4)^{hyp}}$ be the square-tiled surfaces with “rich monodromy” constructed in our paper (see also this post), i.e., the square-tiled surfaces below

$M_{\ast}\in\mathcal{H}(4)^{odd}$

$M_{\ast\ast}\in\mathcal{H}(4)^{hyp}$

associated to the pairs of permutations ${h_{\ast}=(1)(2,3)(4,5,6)}$ and ${v_{\ast}=(1,4,2)(3,5)(6)}$, and ${h_{\ast\ast}=(1)(2,3)(4,5,6)}$ and ${v_{\ast\ast}=(1,2)(3,4)(5)}$.

Remark 2 Here, as it is usual in this theory, we are constructing square-tiled surfaces from a pair of permutations ${h, v\in S_N}$ on ${N}$ elements by taking ${N}$ unit squares and gluing them (by translations) so that ${h(i)}$ is the square to the right of the square ${i}$ and ${v(i)}$ is the square on the top of the square ${i}$.

Proposition 1 For each ${d\geq 3}$ odd, there exists ${M_{\ast}(d)}$ a finite branched cover of ${M_{\ast}}$ of degree ${d}$ in the odd connected component ${\mathcal{H}(5d-1)^{odd}}$ of the minimal stratum ${\mathcal{H}(5d-1)}$ (of Abelian differentials with a single zero of order ${5d-1}$).

Also, there exists ${M_{\ast\ast}(3)}$ a finite branched cover of ${M_{\ast\ast}}$ of degree ${3}$ in the hyperelliptic connected component ${\mathcal{H}(14)^{hyp}}$ of the minimal stratum ${\mathcal{H}(14)}$ (of Abelian differentials with a single zero of order ${14}$).

Finally, for each ${d\geq 5}$, there exists ${M_{\ast\ast}(d)}$ a finite branched cover of ${M_{\ast\ast}}$ of degree ${d}$ in the even connected component ${\mathcal{H}(5d-1)^{even}}$ of the minimal stratum ${\mathcal{H}(5d-1)}$ (of Abelian differentials with a single zero of order ${5d-1}$).

Remark 3 In this statement, the nomenclature “hyperelliptic”, “even” and “odd” refers to the invariants introduced by Kontsevich and Zorich to distinguish between the connected components of strata. We will briefly review these notions below (as we will need them to prove Proposition 1.

We will dedicate the remainder of this post to outline the construction of the translation surfaces ${M_{\ast}(d)}$ and ${M_{\ast\ast}(d)}$ satisfying the conclusions of Proposition 1. In particular, we will divide the discussion into three sections: in the next one, we will quickly review the invariants introduced by Kontsevich-Zorich to classify connected components of strata, and in the last two sections we will construct ${M_{\ast}(d)}$ and ${M_{\ast\ast}(d)}$ respectively.

Posted by: matheuscmss | July 31, 2013

## Furstenberg’s theorem on the Poisson boundaries of lattices of SL(n,R) (part II)

Last time we introduced Poisson boundaries hoping to use them to distinguish between lattices of ${SL(2,\mathbb{R})}$ and ${SL(n,\mathbb{R})}$, ${n\geq 3}$. More precisely, we followed Furstenberg to define and construct Poisson boundaries as a tool that would allow to prove the following statement:

Theorem 1 (Furstenberg (1967)) A lattice of ${SL(2,\mathbb{R})}$ can’t be realized as a lattice in ${SL(n,\mathbb{R})}$ for ${n\geq 3}$ (or, in the language introduced in the previous post, ${SL(n,\mathbb{R})}$, ${n\geq 3}$, can’t envelope a discrete group enveloped by ${SL(2,\mathbb{R})}$).

Here, we recall that, very roughly speaking, these Poisson boundaries ${(B,\nu)}$ were certain “maximal” topological objects attached to locally compact groups with probability measures ${(G,\mu)}$ in such a way that the points in the boundary ${B}$ were almost sure limits of ${\mu}$-random walks on ${G}$ and the probability measure ${\nu}$ was a ${\mu}$-stationary measure giving the distribution at which ${\mu}$-random walks hit the boundary.

For this second (final) post, we will discuss (below the fold) some examples of Poisson boundaries and, after that, we will sketch the proof of Theorem 1.

Remark 1 The basic references for this post are the same ones from the previous post, namely, Furstenberg’s survey, his original articles and A. Furman’s survey.

Posted by: matheuscmss | July 21, 2013

## Furstenberg’s theorem on the Poisson boundaries of lattices of SL(n,R) (part I)

In this previous blog post here (about this preprint joint with Alex Eskin), it was mentioned that the simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle over Teichmüller curves in moduli spaces of Abelian differentials (translation surfaces) can be determined by looking at the group of matrices coming from the associated monodromy representation thanks to a profound theorem of H. Furstenberg on the so-called Poisson boundary of certain homogenous spaces.

In particular, this meant that, in the case of Teichmüller curves, the study of Lyapunov exponents can be performed without the construction of any particular coding (combinatorial model) of the geodesic flow, a technical difficulty occurred in previous papers dedicated to the simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle (such as these articles here and here).

Of course, I was happy to use Furstenberg’s result as a black-box by the time Alex Eskin and I were writing our preprint, but I must confess that I was always curious to understand how Furstenberg’s theorem works. In fact, my curiosity grew even more when I discovered that Furstenberg wrote a survey article (of 63 pages) on this subject, but, nevertheless, this survey was not easily accessible on the internet. For this reason, after consulting a copy of Furstenberg’s survey at Institut Henri Poincaré (IHP) library, I was impressed by the high quality of the material (as expected) and I decided to buy the book containing this survey.

As the reader can imagine, I learned several theorems by reading Furstenberg’s survey and, for this reason, I thought that it could be a good idea to describe here the proof of a particular case of Furstenberg’s theorem on the Poisson boundary of lattices of ${SL(n,\mathbb{R})}$ (mostly for my own benefit, but also because Furstenberg’s survey is not easy to find online to the best of my knowledge).

For the sake of exposition, I will divide the discussion of Furstenberg’s survey into two posts, using Furstenberg’s survey, his original articles and A. Furman’s survey as basic references.

For this first (introductory) post, we will discuss (below the fold) some of the motivations behind Furstenberg’s investigation of Poisson boundaries of lattices of Lie groups and we will construct such boundaries for arbitrary (locally compact) groups equipped with probability measures.

Posted by: matheuscmss | July 6, 2013

## A remark on the Jacobian determinant PDE

A few days ago I crossed by chance the book “The pullback equation for differential forms” of G. Csató, B. Dacorogna and O. Kneuss. Very roughly speaking, this book concerns the existence and regularity of solutions to the partial differential equation (PDE)

$\displaystyle \varphi^*(\eta_0)=\eta_1$

where ${\eta_0}$ and ${\eta_1}$ are given ${k}$-forms and ${\varphi}$ is an unknown map.

As it turns out, this is a non-linear (for ${k\geq 2}$) homogenous (of degree ${k}$ in the derivatives of ${\varphi}$) of first-order system of ${\binom{n}{k}}$ PDEs on ${\varphi}$.

The first time I got interested in this (pullback) PDE was some years ago because of its connection with the question of constructing “nice” perturbations of volume-preserving dynamical systems (see this paper here [and its corrigendum here]).

More concretely, suppose that you are studying the dynamical features of volume-preserving diffeomorphism ${f}$ and you want to test whether ${f}$ has some given robust property ${\mathcal{P}}$, e.g., ${f}$ is robustly transitive (i.e., all volume-preserving diffeomorphisms ${g}$ obtained from small perturbations of ${f}$ have some dense orbit). Then, you can use the pullback equation to contradict this robust property for ${f}$ along the following lines.

Assume that you found some region ${U}$ of the phase space where ${f}$ is “strangely” close to a local volume-preserving dynamical system ${g}$ violating the property ${\mathcal{P}}$ (e.g., ${g}$ is not transitive because it leaves invariant some open subset ${V\subset U}$). Hence, you could show that ${f}$ doesn’t satisfy ${\mathcal{P}}$ in a robust way if you can “glue” ${f}$ and ${g}$ in a conservative way to obtain a volume-preserving diffeomorphism ${h}$ behaving as ${g}$ inside ${U}$ and behaving as ${f}$ outside some neighborhood of ${\overline{U}}$. Since the local dynamics ${g}$ violates ${\mathcal{P}}$, we have that the perturbation ${h}$ of ${f}$ also violates ${\mathcal{P}}$, so that the robust property ${\mathcal{P}}$ is not verified by ${f}$.

Here, the pullback equation for volume forms says that you can glue ${g}$ and ${f}$ in a volume-preserving way once you glued them in some non-volume-preserving way: indeed, let ${h_0}$ be a possibly non-volume preserving diffeomorphism gluing ${f}$ and ${g}$; since the pullback equation for volume forms ${\omega_0}$ and ${\omega_1}$ translates into the Jacobian determinant PDE ${\det D\varphi(x)= \theta(x)}$ where ${\theta}$ is a density function (corresponding to ${\omega_1/\omega_0}$), by letting ${\varphi_0}$ be a solution of the Jacobian determinant equation with ${\theta(x)=\det(Dh_0(h_0^{-1}(x)))}$, we can “correct” the “defect” of ${h_0}$ to preserve volume by replacing ${h_0}$ by the volume-preserving map ${\varphi_0\circ h_0}$.

Historically speaking, the Jacobian determinant equation (i.e., the pullback equation for volume forms) was discussed in details in this paper of B. Dacorogna and J. Moser (from 1990), and some of the results in this paper were recently used to perform “nice” perturbations/regularizations of volume-preserving dynamical systems (see, e.g., this paper here of A. Avila and the references therein).

An interesting technical point about the paper of B. Dacorogna and J. Moser is that they provide two results about the Jacobian determinant equation (stated below only for domains in ${\mathbb{R}^n}$ for sake of simplicity of the exposition):

• (a) given a positive density ${\theta\in C^{k+\alpha}}$ (where ${k\in\mathbb{N}}$ and ${0<\alpha<1}$) on the closure ${\overline{\Omega}}$ of an open bounded smooth connected domain ${\Omega\subset\mathbb{R}^n}$ with total mass ${\int_{\Omega}\theta=1}$, there exists a ${C^{k+1+\alpha}}$ diffeomorphism ${\varphi}$ such that ${\det D\varphi=\theta}$ on ${\Omega}$ and ${\varphi(x)=x}$ on ${\partial\Omega}$;
• (b) given a positive density ${\theta\in C^k}$ (where ${k\in\mathbb{R}}$, ${k\geq 1}$), on the closure ${\overline{\Omega}}$ of an open bounded smooth connected domain ${\Omega\subset\mathbb{R}^n}$ with total mass ${\int_{\Omega}\theta=1}$, there exists a ${C^{k}}$ diffeomorphism ${\varphi}$ such that ${\det D\varphi=\theta}$ and ${\varphi(x)=x}$; furthermore, if ${\textrm{supp}(\theta-1)\subset\Omega}$ (i.e., the density ${\theta}$ equals ${1}$ near the boundary ${\partial \Omega}$), then ${\varphi}$ can be chosen so that ${\textrm{supp}(\varphi-id)\subset\Omega}$ (i.e., ${\varphi}$ is the identity diffeomorphism near ${\partial\Omega}$).

In a nutshell, the result in item (a) is proven in (Section 2 of) Dacorogna-Moser’s paper via global methods based on elliptic regularity of the Laplacian operator, while the result in item (b) is proven in (Section 3 of) Dacorogna-Moser’s paper via local methods based on reduction to ordinary differential equations (ODE’s).

As the attentive reader noticed, there is a tradeoff in the results in items (a) and (b): together, these results say that, if you want to prescribe a density while controlling the behavior of the diffeomorphism near the boundary ${\partial\Omega}$, then you can do it via item (b) at the cost that you will get no gain of regularity in the sense that the density and the diffeomorphism have the same regularity ${C^k}$; on the other hand, if you want a gain of regularity, then you can do it via item (a) at the cost of giving up on the control of the diffeomorphism near ${\partial\Omega}$ (the best you can ensure is that the boundary is pointwise fixed but you don’t know that the diffeomorphism is the identity near ${\partial\Omega}$).

Remark 1 In some sense, the loss of control on the behavior of the diffeomorphism in item (a) is essentially due to the fact that Dacorogna and Moser construct their diffeomorphisms ${\varphi}$ using the solutions of the elliptic PDE ${\Delta a = \theta - 1}$ (with Neumann boundary condition say) because ${\Delta = \textrm{div}\,\textrm{grad}}$ and ${\textrm{div}(v) = \theta-1}$ is the linearized equation associated to the Jacobian determinant PDE. Indeed, since the solutions of ${\Delta a = \theta - 1}$ are obtained by convolution of ${\theta-1}$ and the Poisson kernel, the fact that ${\theta-1=0}$ near ${\partial\Omega}$ doesn’t imply that ${a=0}$ near ${\partial\Omega}$ (in other words, the convolution is a global operation and thus the local behavior of ${\theta-1}$ is not sufficient to control the local behavior of ${a}$). In particular, given that the behavior of the diffeomorphisms ${\varphi}$ of Dacorogna-Moser near ${\partial\Omega}$ are driven by the behavior of ${a}$ (or rather ${\textrm{grad}(a)}$) near ${\partial\Omega}$, we see that the gain regularity in item (a) above comes with the cost of loss of control of ${\textrm{supp}(\varphi-id)}$.

For the applications of Dacorogna-Moser’s theorems in Dynamical Systems so far (e.g., Avila’s theorem on the regularization of volume-preserving diffeomorphisms), the control of ${\textrm{supp}(\varphi-id)}$ is more relevant than the gain of regularity and this explains why item (b) is more often used in dynamical contexts than item (a).

Given this scenario, it is natural to ask whether one can have the best from both worlds (or items), i.e., gain of regularity and control of support of diffeomorphisms with prescribed Jacobian determinant.

Of course, as it was pointed out by B. Dacorogna and J. Moser themselves in their 1990 paper (see item (iv) at page 14 of their article), one needs a new ingredient to attack this question. So, after stumbling upon the book of Csató, Dacorogna and Kneuss from 2012, I thought that there might be news on this question since the last time I have looked at it.

However, I saw some “bad” news by page 18-19 of Csató-Dacorogna-Kneuss book, where they say: “In Section 10.5 (cf. Theorem 10.11), we present a different approach proposed by Dacorogna and Moser [33] to solve our problem. This method is constructive and does not use the regularity of elliptic differential operators; in this sense, it is more elementary. The drawback is that it does not provide any gain of regularity, which is the strong point of the above theorem. However, the advantage is that it is much more flexible. For example, if we assume in (1.2) [a variant of Jacobian determinant equation] that

$\displaystyle \textrm{supp}(f-g)\subset\Omega,$

then we will be able to find ${\phi}$ such that

$\displaystyle \textrm{supp}(\phi-id)\subset\Omega.$

This type of result, unreachable by the method of elliptic partial differential equations, will turn out to be crucial in Chapter 11.”

Nevertheless, the book of Csató-Dacorogna-Kneuss brought also some good news: there were some progress on the question of gain of regularity in Poincaré lemma, that is, the question of writing a given closed cohomologically trivial form as an exact form (i.e., the differential of another form). After reading about this advance on regularity in Poincaré lemma, I noticed that this is precisely what one needs to modify Dacorogna-Moser’s arguments in order to get gain of regularity and control of the support thanks to a “correction of support argument” that I first read in this paper of Avila here.

In summary, it is possible to gain regularity and control the support in the Jacobian determinant equation by combining the arguments of Dacorogna-Moser and the Poincaré lemma in Csató-Dacorogna-Kneuss book. Evidently, this result per se is not publication-quality (it is just a combination of important results by other authors), but I thought that it could be a nice idea to leave some trace of this fact in this blog in case someone needs this information in the future.

So, the main goal of today’s post is the existence of solutions of the Jacobian determinant equation whose support are controlled and with a gain of regularity.

For sake of simplicity of the exposition, we will simplify our setting by considering the following particular situation (appearing naturally in some dynamical applications of the Jacobian determinant equation): ${\Omega=B_2(0)-B_{1/4}(0)}$ where ${B_r(0)}$ denotes the open ball of radius ${r}$ centered at the origin ${0\in\mathbb{R}^n}$ and ${\theta:\overline{\Omega}\rightarrow\mathbb{R}}$ is a positive (density) function of class ${C^{k+\alpha}}$ for some ${k\in\mathbb{N}}$ and ${0<\alpha<1}$ such that ${\theta\equiv 1}$ on the compact neighborhood ${U=\overline{B_2(0)-B_1(0)}\cup \overline{B_{1/2}(0)-B_{1/4}(0)}}$ of ${\partial\Omega}$. In this context, we will show that:

Proposition 1 Given ${0<\gamma<\alpha<1}$, there exists a constant ${C=C(\Omega, k, \alpha, \gamma)}$ such that, if ${\int_{\Omega}\theta=1}$, the Jacobian determinant equation

$\displaystyle \det D\varphi(x)=\theta(x)$

has a solution ${\varphi\in\textrm{Diff}^{k+1+\alpha}(\overline{\Omega})}$ with ${\varphi\equiv id}$ on ${U}$ and ${\|\varphi-id\|_{C^{k+1+\gamma}}\leq C \|\theta-1\|_{C^{k+\alpha}}}$.

Remark 2 As we will see, even though Proposition 1 implies item (a), a theorem of Dacorogna-Moser, the proof of Proposition 1 uses the results of Dacorogna and Moser.

As we already mentioned, the proof of this proposition is a modification of the arguments of Dacorogna-Moser using the Poincaré lemma of Csató-Dacorogna-Kneuss. For this reason, we will divide this post into two sections: the next one serves to recall some preliminary results (mostly consisting of modifications of some results in Section 2 of Dacorogna-Moser’s paper) and in the short final section we will complete the proof of Proposition 1.

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)