Today we will discuss the 3rd lecture (last Wednesday, Jan. 25, 2012) by J.-C. Yoccoz (corresponding to his 2011-2012 course on square-tiled surfaces), but before doing so, we will make a quick review of the material of this previous post.

Let be an origami associated to a finite group generated by two elements and , and a subgroup of containing no nontrivial normal subgroup. Denoting by the normalizer of in , we have that the automorphism group is . For the sake of the exposition, we will denote .

In this language, we saw that the absolute homology group has a natural -invariant decomposition as

where has dimension and has codimension (i.e., dimension ).

Finally, we computed an explicit formula for the multiplicity in of a -irreducible -representation of character , and we used this formula to prove that (for any origami and any ).

So, this is it as far as the quick revision is concerned. Below the fold the reader will find my notes for J.-C. Yoccoz 3rd lecture: his main goal in it was to show how to use the information (derived in the previous two lectures) about -representations to deduce useful facts about -representations. As it turns out, a significant part of this classical and, in particular, the representation theory facts we’re going to use (without proof) can be found in the books of J.-P. Serre and W. Fulton and J. Harris. Nevertheless, this lecture will contain some new facts (from the forthcoming paper by J.-C. Yoccoz, D. Zmiaikou and C.M.) concerning the specific case of representations related to origamis.

In this previous post (about J.-C. Yoccoz 2011-2012 course at College de France), we reviewed the distinct points of view on origamis, and we started the discussion of the action of the automorphism group on the first absolute homology group of the origami. We quickly recall the main points of the previous post for the reader’s convenience.

Given an origami , we saw how to canonically describe it (through its monodromy group) by a finite group generated by two elements and , and a choice of subgroup such that , so that one has . Also, and act on the set of squares of by and . Furthermore, denoting by the normalizer of in , one has acting as , for .

Next, given a subfield , we decomposed and we reduced the problem of studying the action of on to understanding the action of on , and, to do so, we considered the set , and we proved that

as -module. In this way, since and , we were led to the study of the action of on . At this stage, we noticed that is the set of orbits of acting to the right on or equivalently the set of orbits of acting on by .

Nevertheless, we introduced the following notation: is the point of associated to , is the stabilizer of for the action of on , and is the order of the commutator .

Finally, if we denote by the smallest integer such that , we showed (in the very end of the previous post) the following lemma:

Lemma. .

After this quick revision, we will enter (below the fold) the content of Yoccoz’s 2nd lecture (on January 18, 2012).

From January 11 to March 21, Jean-Christophe Yoccoz delivers (on Wednesdays) his course (of academic year 2011-2012) at Collège de France. As the reader can find in his webpage, he decided to make a continuation of his last course (about square-tiled surfaces) and so he entitled the current series of lectures “Surfaces à petits carreaux (suite)”.

After following the first two lectures, I thought it could be a nice idea to try to make available the notes I’m taking for this course. So, I plan to write a series of posts whose titles will have the form “SPCS x” (where SPCS stands for“Surfaces à petits carreaux (suite)” and “x” stands for the number of the lecture). Of course, this goes without saying that any errors and/or mistakes are surely my sole fault: indeed, since the course is delivered in French, it may happen that I misinterpret some points.

Below the fold the reader will find my first set of notes, i.e., SPCS 1, corresponding to Yoccoz’s lecture on last January 11th.

In two recent papers, E. Lanneau and D.-M. Nguyen, and M. Möller studied Teichmüller curves in genera and steaming from Prym eigenforms. Their work can be seen as a sort of “follow-up” to C. McMullen’s seminal works (who completely treated these objects in genus ), and, from the point of view of Dynamical Systems, they are very interesting as a source of examples where the Lyapunov exponents of the Kontsevich-Zorich cocycle can be “described” (see, e.g., these links here for an introduction to the ergodic theory of the Kontsevich-Zorich cocycle). For instance, as it was noticed by M. Möller, C. Weiss and myself (independently), it is really easy to put together the works of D. Chen and M. Möller, A. Eskin, M. Kontsevich and A. Zorich, and M. Möller to compute the Lyapunov spectrum of these Teichmüller curves.

Of course, the knowledge of Lyapunov exponents per se may not seem very exciting, but during the Christmas Workshop 2011 of Karlsruhe University, C. Weiss gave a talk (about his PhD thesis under the supervision of M. Möller) showing how this information about Lyapunov exponents can be put forward to exhibit new special curves (i.e., Kobayashi geodesics) inside Hilbert modular surfaces. As it turns out, special curves in Hilbert modular surfaces are very rigid objects, and before C. Weiss’ result, the list of previously known special curves was not very long: it contained Hirzebruch-Zagier cycles (a.k.a. Shimura curves or twisted diagonals) and twisted Teichmüller curves solely.

The goal of today’s post is to revisit the construction of Teichmüller curves through Prym eigenforms and to give a (very rough) sketch of proof of C. Weiss’ theorem.

In the previous post we proved Szemerédi’s regularity lemma, and now we give a few of its various applications: the triangle removal lemma, Roth’s theorem on the existence of arithmetic progressions of length in subsets of the integers with positive density, the corner’s theorem and, finally, the graph removal lemma.

1. Triangle removal lemma

Most applications of Szemerédi’s regularity lemma deal with monotone problems, when throwing in more edges can only help. In these applications, one starts applying the original form of the regularity lemma to create a regular partition, then gets rid of all edges within the clusters of the partition, also the edges of non-regular pairs as well as those of regular pairs with small density. The leftover “pure” graph is much easier to handle and still contains most of the original edges. This is what happens in proving the triangle removal lemma.

The triangle removal lemma is the (intuitive, yet nontrivial) fact that if one has to delete at least edges of a graph with vertices to destroy all triangles in it, then the graph must contain at least triangles, where . If one only thinks naively, the conclusion is that the graph contains at least triangles, and the strength of the triangle removal lemma is that, instead of quadratic, the number of triangles is cubic. It was first proved by Ruzsa and Szemerédi in the paper Triple systems with no six points carrying three triangles, who also observed it implies Roth’s theorem, as we shall see in the next section.

Szemerédi’s regularity lemma is an important tool in discrete mathematics, specially in graph theory and additive combinatorics. It says that, in some sense, all graphs can be approximated by random-looking graphs. The lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. One of its applications is the triangle removal lemma which, as observed by Ruzsa and Szemerédi in the paper Triple systems with no six points carrying three triangles, gives a proof of Roth’s theorem on the existence of arithmetic progressions of length in subsets of the integers with positive density (see ERT13 for an ergodic theoretical proof).

These days I gave two talks (the first one on Tuesday, December 13, entitled Some examples of cocycles with wild central Oseledets bundle during the conference Recent advances in modern dynamics held at the University of Warwick, and the second one on Wednesday, December 21, entitled Neutral Oseledets bundles of the Kontsevich-Zorich cocycle during the Christmas workshop of Karlsruhe University) around some results (from joint works with G. Forni and A. Zorich, and A. Avila and J.-C. Yoccoz) on the neutral Oseledets bundles of the Kontsevich-Zorich cocycle partly announced in this previous post here. Below the fold, the reader will find an expanded version of my lecture notes.

Acknowledgments. I would like to thank the organizers of the two conferences above (in particular, Corinna Ulcigrai and Gabriela Schmithüsen) for the invitation to deliver the talks at the origin of these notes.

Last Friday G. Forni, A. Zorich and myself uploaded to ArXiv the article Lyapunov spectrum of invariant subbundles of Hodge bundle. In this paper (partly announced here), we study the behavior of Kontsevich-Zorich cocycle restricted to (Teichmuller flow and/or ) equivariant subbundles of the Hodge bundle.

Firstly, let me say that we mostly consider this article as a survey. Indeed, a large portion of this article is dedicated to revisit some variational formulas in G. Forni’s paper using the perspective of Differential Geometry: more precisely, we re-interpret Forni’s variational formulas for the growth of Hodge norms of vectors and isotropic subspaces of the Hodge bundle in terms of features of the second fundamental form (a.k.a. Kodaira-Spencer map) and the curvature form of the Gauss-Manin connection on the Hodge bundle.

Then, we use this point of view to detect the minimal set of assumptions on a subbundle of the (real) Hodge bundle under which a “Kontsevich-Zorich-Forni like formula” for the sum of Lyapunov exponents holds: whenever is a -equivariant and Hodge-star invariant, the sum of the non-negative Lyapunov exponents related to is described by the average of the eigenvalues of a certain “curvature form” (see Corollary 3.5).

Furthermore, also by using this point of view, we deduce a new result (Theorem 3) relating the neutral Oseledets bundle of the Kontsevich-Zorich cocycle (i.e., the Oseledets subspace associated to the zero Lyapunov exponents) and the annihilator of the second fundamental form of the Gauss-Manin connection. In particular, we show that, if is Teichmuller-flow invariant then , and, if is -invariant, then , and hence and coincide whenever they are -invariant, i.e., the annihilator of the second fundamental form is a natural candidate for the neutral Oseledets bundles of the Kontsevich-Zorich cocycle (at least under the appropriate invariance assumptions). An interesting corollary of this (and the fact that the infinitesimal variation of the Hodge norm along the Kontsevich-Zorich cocycle is measured by the second funamental form, i.e., , see Lemma 2.4) is the fact that the Kontsevich-Zorich cocycle acts by isometries (with respect to the Hodge norm) along whenever this subbundle is -invariant.

Finally, we “test” this new result against two classes of examples (presented in Appendix A and B). In the first class of examples, namely, square-tiled cyclic covers, we verify that both and are -invariant. As a consequence, we derive that (see Theorem 7). In other words, the neutral Oseledets bundle (responsible for the zero exponents of the Kontsevich-Zorich cocycle) has a natural geometric explanation: it is the annihilator of the second fundamental form.  Some nice consequences of this fact are that the Kontsevich-Zorich cocycle acts by isometries along , and is continuous (and actually real-analytic) in the case of square-tiled cyclic covers: indeed, this is so because the second fundamental form is a continuous (actually, real-analytic) object. However, as we announce  in Appendix B (leaving the details for a forthcoming paper), the neutral Oseledets bundle doesn’t coincide with in general! In fact, based on some constructions of C. McMullen, we exhibit an example where is not -invariant (and hence ) despite the fact that and have the same rank! In any event, even though the annihilator of the second fundamental form is not responsible for the zero exponents in this example, the mechanism for the existence of is not very complicated: essentially, we are dealing with a cocycle of matrices preserving an indefinite non-degenerate Hermitian form (i.e., we have a cocycle of matrices where is the signature of the invariant Hermitian form), and a simple (linear-algebra) argument allows to prove the existence of zero exponents in this context. Also, let me point out that in this example the Kontsevich-Zorich cocycle also acts by isometries, but the conceptual reason behind this is different from the square-tiled cyclic covers case! Of course, we will come back to this issue later in this blog (most likely when the promised forthcoming paper comes out).

Closing this post, let me make two points. The first one is that, besides the “applications” given in the Appendices to this survey article, we feel that this point of view (of using the second fundamental form to understand the Kontsevich-Zorich cocycle) might be helpful in other contexts (and that’s what motivated us to write down this survey). For instance, Alex Eskin communicated to us that the discussion in our survey is useful when trying to derive certain semisimplicity statements of the “algebraic hulls” (in the sense of Zimmer) of the Kontsevich-Zorich cocycle (needed in his work with Maryam Mirzakhani on classification of -invariant measures in moduli spaces). The second point is that, while in the case of square-tiled cyclic covers we showed that the neutral Oseledets bundle is continuous by comparison with the annihilator of the second fundamental form, the same kind of reasoning can’t be applied to the second class of examples (in Appendix B of our article), and so it is natural to ask about the regularity of in this situation. Here, in a work (in progress) by A. Avila, J.-C. Yoccoz and myself, we are able to show (among other things) that is not continuous at all, so that it is only measurable at best. In particular, this gives an example of a symplectic cocycle whose neutral Oseledets bundle is not continuous (in contrast with “most” examples in the literature where zero Lyapunov exponents “usually” are associated to continuous subbundles).  Evidently, I also plan to come more on this work in progress in due time, but for now I think that’s all I have to say on zero exponents of the Kontsevich-Zorich cocycle!

A few weeks ago, I was invited by my friend Jairo Bochi to give a “general audience like” talk (that I’ll deliver today) at UFRJ (Brazil) in a Dynamical Systems seminar called EDAI. After thinking a bit, I decided to discuss a recent beautiful theorem of V. Delecroix, P. Hubert and S. Lelièvre on the diffusion rates for the Ehrenfest wind-tree model of Lorenz gases. Here, my choice of theme was motivated by the following two facts:

• this theorem has some roots in Physics (more precisely, Statistical Mechanics) and
• its proof has a lot to do with recent advances in the study of the Lyapunov exponents of the Kontsevich-Zorich cocycle (a subject that I’m particularly interested in).

So, while I was preparing these slides here (written in Portuguese), I thought that it could be a nice idea to publish a sort of “extended version” of the slides in this blog. The outcome of this is the text below the fold. I hope you’ll enjoy your reading as much as I enjoyed preparing these notes!

Below I’m reproducing a message from Artur Avila about some available post-doctoral positions:

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Postdoctoral Fellowship for the academic year 2012-2013

Postdoctoral fellowships at the Institut de Mathematiques de Jussieu
in Paris, for 12 months (extensible up to 24 months), starting on
September 1st 2012, are offered as part of the ongoing E.R.C. project
“Quasiperiodic”. There is a possibility of starting earlier.

Candidates should be finishing or have recently finished their Ph.D
and work in quasiperiodic dynamics with special emphasis on
quasiperiodic cocycles and Schrödinger operators, interval exchange
transformation and Teichmüller flows.

The applications should be sent by e-mail to Artur Avila
(artur@math.jussieu.fr) and consist of a CV (+publication list), a
short description of the past research of the candidate and a research
project.  The deadline for application is February 1st 2011.

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