This memoir is a preliminary step towards obtaining a HDR diploma (a general requirement in the French academic system to supervise PhD students, etc.) and it summarizes some of my researches after my PhD thesis (or, more specifically, my researches on the dynamics of the Teichmüller flow).

Despite the title and abstract in French, the main part of this memoir is in English.

The first chapter of the memoir recalls many basic facts on the action on the moduli spaces of translation surfaces. In particular, this chapter is a general introduction to all subsequent chapters of the memoir. Here, the exposition is inspired from Zorich’s survey, Yoccoz’s survey and our survey with Forni.

After reading the first chapter, the reader is free to decide the order in which the remainder of the memoir will be read: indeed, the subsequent chapters are independent from each other.

The second chapter of the memoir is dedicated to the main result in my paper with Avila and Yoccoz on the Eskin-Kontsevich-Zorich regularity conjecture. Of course, the content of this chapter borrows a lot from my four blog posts on this subject.

The third chapter of the memoir discusses my paper with Schmithüsen on complementary series (and small spectral gap) for explicit families of arithmetic Teichmüller curves (i.e., -orbits of square-tiled surfaces).

The fourth chapter of the memoir is dedicated to my paper with Wright on the applications of the notion of Hodge-Teichmüller planes to the question of classification of algebraically primitive Teichmüller curves. Evidently, some portions of this chapter are inspired by my blog posts on this topic.

The fifth chapter of the memoir is consacrated to Lyapunov exponents of the Kontsevich-Zorich cocycle over arithmetic Teichmüller curves. Indeed, after explaining the results in my paper with Eskin about the applications of Furstenberg boundaries to the simplicity of Lyapunov exponents, we spend a large portion of the chapter discussing the Galois-theoretical criterion for simplicity of Lyapunov exponents developed in my paper with Möller and Yoccoz. Finally, we conclude this chapter with an application of these results (obtained together with Delecroix in this paper here) to a counter-example to a conjecture of Forni.

The last chapter of the memoir is based on my paper with Filip and Forni on the construction of examples of `exotic’ Kontsevich-Zorich monodromy groups and it is essentially a slightly modified version of this blog post here.

Closing this short post, let me notice that the current version of this memoir still has no acknowledgements (except for a dedicatory to Jean-Christophe Yoccoz’s memory) because I plan to add them only after I get the referee reports. Logically, once I get the feedback from the referees, I’m surely going to include acknowledgements to my friends/coauthors who made this memoir possible!

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In what follows, I’m reproducing my notes for Jean-François Quint’s lecture. (As usual, all errors/mistakes in the sequel are my responsibility.)

**1. Introduction **

** 1.1. Limit sets of semigroups of matrices **

Let be a semigroup of invertible real matrices.

Recall that:

- is
*irreducible*if there are no non-trivial -invariant subspaces, i.e., and imply or ; - is
*proximal*if it contains a proximal element , i.e., has an unique eigenvalue with maximal modulus which has multiplicity one in the characteristic polynomial of ; equivalently, , , and has spectral radius or, in other terms, the action of on the projective space has an attracting fixed point.

Proposition 1Let be a irreducible and proximal semigroup. Then, the action of on admits a smallest non-empty invariant closed subset called the limit set of .

*Proof:* Let . It is clear that is non-empty, closed and invariant. Moreover, is the smallest subset with these properties thanks to the following argument. Let be a proximal element. If , then converges to as . If , we use the irreducibility of to find an element such that and, *a fortiori*, converges to as .

** 1.2. Stationary measures **

Suppose that is a probability measure on a semigroup acting on a space . We say that a probability measure on is –*stationary* if it is -invariant on average, i.e.,

is equal to .

In the case of irreducible and proximal semigroups of matrices, the following theorem of Furstenberg and Kesten ensures the existence and uniqueness of stationary measures for the corresponding projective actions:

Theorem 2 (Furstenberg-Kesten)Let be a Borel probability measure on and denote by the subsemigroup generated by the elements in the support of . Suppose that is irreducible and proximal. Then, has an unique -stationary measure on and .

In what follows, we shall also assume that is *strongly irreducible*, i.e., for all non-trivial proper subspaces , and we will be interested in the nature of in Furstenberg-Kesten theorem.

It is possible to show that if is absolutely continuous with respect to the Lebesgue (Haar) measure (on ), then is absolutely continuous with respect to the Lebesgue measure (on ).

For this reason, we shall focus in the sequel on the following question:

Can be absolutely continuous when is *finitely supported*?

It was shown by Kaimanovich and Le Prince that the answer to this question is *not* always positive:

Theorem 3 (Kaimanovich-Le Prince)There exists finite (actually, ) such that spans a Zariski dense subsemigroup of , but is the support of a probability measure such that the associated stationary measure on is singular with respect to the Lebesgue measure.

On the other hand, Bárány-Pollicott-Simon and Bourgain showed that the answer to this question is sometimes positive:

Theorem 4 (Bárány-Pollicott-Simon, Bourgain)There exists finite supporting a probability measure such that the corresponding stationary measure is absolutely continuous with respect to Lebesgue.

Remark 1As it was pointed out by Quint, the examples produced by Bourgain are explicit, but it would be desirable to get simpler explicit examples of sets satisfying the previous theorem. In this direction, he asked the following question. Denote by , and , , and consider the probability measures

Is it true that, for each fixed , if is small enough (and typical?), then the stationary measure associated to is absolutely continuous with respect to the Lebesgue measure? (Note that if is very large, then we are in the regime described by Kaimanovich-Le Prince theorem 3.)

** 1.3. Statement of the main result **

In a recent paper, Benoist and Quint extended Theorem 4 to higher dimensions:

Theorem 5 (Benoist-Quint)For any , there exists finite and a probability measure with and proximal and strongly irreducible such that the corresponding stationary measure on is absolutely continuous with respect to Lebesgue.

The remainder of this post is dedicated to the proof of this result.

**2. Proof of the main theorem **

** 2.1. Spectral theory of quasi-compact operators **

Let be a Banach space and denote by the space of bounded linear operators on .

Given , recall that the compact non-empty set

is the *spectrum* of , and the quantity

is the *spectral radius* of .

The space of compact operators is an ideal and the quotient comes equipped with a natural norm .

Recall that the *essential spectrum* of is

and the *essential spectral radius* of is

Note that and . Moreover, these objects are the same for and its adjoint :

Proposition 6One has the following identities: , , and .

The next proposition explains that the spectrum and the essential spectrum morally differ only by eigenvalues of finite multiplicity:

Proposition 7If , then there exists a decomposition into closed subspaces such that is finite-dimensional, , , and .

** 2.2. Spectral criterion for absolute continuity **

Let be a Borel probability measure on and consider the natural action of on .

Given a function , let .

Remark 2when is finitely supported.

We equip with the round measure induced from the natural Lebesgue measure on the sphere .

If has compact support, then is a bounded operator on .

One can infer the absolute continuity of the stationary measure of from the spectral properties of thanks to the following proposition:

Proposition 8If is proximal and strongly irreducible, and , then the -stationary measure on is absolutely continuous with respect to , i.e., .

*Proof:* Note that (where is the constant function with value one), so that .

By hypothesis, . Thus, there exists with . By definition, this means that the absolutely continuous measure is the -stationary measure.

** 2.3. Application of the spectral criterion **

The result in Theorem 5 (i.e., the case ) is easier to derive than Theorem 4 (i.e., the case ) because has elements equidistributing very quickly. Here, the word equidistribution means the following: if is a probability measure on , is Zariski dense on , then we say that the elements in the support of equidistribute whenever for all (with standing for the Haar measure).

This equidistribution property holds in presence of *spectral gap*, i.e., (where is the subspace of -functions with zero average [for Haar measure]). In particular, the works of Drinfeld and Margulis provide examples of elements of equidistributing very quickly:

Theorem 9 (Margulis and Drinfeld)There exists finite and a probability measure with and .

At this point, the proof of Theorem 5 is almost over. Indeed, if we take a proximal element and we denote by the probability measure provided by Margulis and Drinfeld, then the sequence of measures

have the property that converges to a rank one operator. Therefore,

In particular, by Proposition 8, it follows that satisfies the conclusions of Benoist-Quint theorem 5 for any sufficiently large.

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Last week, we spent a couple of days revisiting the content of my blog post on Sarnak’s question about thin KZ monodromies and we realized that the origami of genus 3 discussed in this post turns out to exhibit *arithmetic* KZ monodromy! (In particular, this answers my Mathoverflow question here.)

In this very short post, we show the arithmeticity of the KZ monodromy of .

**1. Description of the KZ monodromy of **

The KZ monodromy of is the subgroup of generated by the matrices

of order three: see Remark 9 in this post here.

For the sake of exposition, we are going to permute the second and fourth vectors of the canonical basis using the permutation matrix

so that the KZ monodromy is the subgroup .

Remark 1This change of basis is purely cosmetical: it makes that the symplectic form preserved by these matrices in is

Denote by the subgroup of unipotent upper triangular matrices in .

**2. Arithmeticity of the KZ monodromy of **

A result of Tits says that a Zariski-dense subgroup such that has finite-index in must be *arithmetic* (i.e., has finite-index in ).

Since we already know that is Zariski-dense (cf. Proposition 3 in this post here), it suffices to check that has finite-index in .

For this sake, we follow the strategy in Section 2 of this paper of Singh and Venkataramana, namely, we study matrices in fixing the first basis vector and, *a fortiori*, stabilizing the flag .

After asking Sage to compute a few elements of (conjugates under of words on , , and of size ) fixing the basis vector , we found the following interesting matrices:

and

In order to check that has finite index in , we observe that

are elements in generating the positive root groups of . In particular, has finite-index in , so that the argument is complete.

Remark 2It is worth noticing that all non-arithmetic Veech surfaces in genus two provide examples of thin KZ monodromy, but this is not the case for origamis (arithmetic Veech surfaces) of genus 2 in the stratum with tiled by squares (as well as for the origami of genus three mentioned above). In particular, this indicates that Sarnak’s question about existence and/or abundance of thin KZ monodromies among origamis might have an interesting answer…

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Such origamis were baptized *regular origamis* by David Zmiaikou in his PhD thesis and the study of the so-called Kontsevich-Zorich over regular (and quasi-regular) origamis was pursued in our joint article with Jean-Christophe Yoccoz.

In this short post, I would like discuss some informal questions posed by Jean-Christophe right after the completion of our joint paper with David Zmiaikou (in order to keep a trace of them in case someone [e.g., myself] wants to think about this matter in the future).

**1. Regular origamis and commutators **

The commutator determines the nature of the conical singularities of the origami : in fact, has exactly such singularities and the total angle around each of them singularities is .

Also, Yoccoz, Zmiaikou and I showed that the “potential blocks” of the Kontsevich-Zorich cocycle over the -orbit of are *completely* determined by the commutator (cf. Theorem 1.1 of our paper).

Given this scenario, Jean-Christophe asked whether the Lyapunov exponents of the Kontsevich-Zorich cocycle of were also completely determined from the knowledge of .

**2. Lyapunov exponents and commutators **

By the time we finished our joint paper, David Zmiaikou pointed out to me that the commutator is not a complete invariant for many algebraic question about : for example, Daniel Stork proved (among other things) that the pairs of permutations and have the same commutator but they generate *distinct* T-systems of the alternate group (cf. Section 2.6 of David Zmiaikou’s PhD thesis for more comments).

This remark led me to play with the origamis and in an attempt to answer Jean-Christophe’s question.

First, note that both of them have conical singularities and the total angle around each of them is . In particular, both and have genus .

On the other hand, a short computation with the aid of Sage (or a long calculation by hand) reveals that the -orbit of has cardinality and the -orbit of has cardinality (this is closely related to the comments in Section 5 of Stork’s paper).

Remark 1Recall that it is easy to algorithmically compute -orbits of origamis described by two permutations and of a finite collection of squares because is generated by and , and these matrices act on pairs of permutations by and (and the permutations and generate the same origami).

Moreover, this calculation also reveals that

- the -orbit of decomposes into four -orbits:
- two -orbits have size and all origamis in these orbits decompose into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height .

- the -orbit of decomposes into three -orbits:
- one -orbit contains a single origami decomposing into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height ;
- one -orbit has size and all origamis in this orbit decompose into horizontal cylinders of width and height .

This information can be plugged into the Eskin-Kontsevich-Zorich formula to determine the sums and of the non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over the -orbits of and .

Indeed, if is an origami with conical singularities whose total angles around them are , , then Alex Eskin, Maxim Kontsevich and Anton Zorich showed that the sum of all non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over is

where is the decomposition of into horizontal cylinders and , resp. is the height, resp. width, of the horizontal cylinder .

In our setting, this formula gives

and

that is,

In particular, this allowed me to answer Jean-Christophe’s informal question: the knowledge of the commutator is *not* sufficient to determine the Lyapunov exponents.

**3. Lyapunov exponents and T-systems? **

Logically, this discussion led Jean-Christophe to ask a new question: how does the Lyapunov exponents of relate to algebraic invariants of ? For example, is the `Lyapunov exponent invariant’ equivalent to `T-systems invariant’ (or is the `Lyapunov exponent invariant’ a completely new invariant? [and, if so, can it be used to derive new interesting algebraic consequences for ?])

Remark 2André Kappes and Martin Möller used Lyapunov exponents in another (but not completely unrelated) setting as `new invariants’ to solve an algebraic problem (namely, classify commensurability classes of non-arithmetic lattices constructed by Pierre Deligne and George Mostow).

(Jean-Christophe and) I plan(ned) to come back to this question at some point in the future, but for now I will close this post here since I have nothing to say about this matter.

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Nevertheless, as my close friends know, Jean-Christophe’s influence on me goes way beyond Mathematics, and, after spending the last month trying to cope with this terrible loss, I believe that today I’m now ready from the emotional point of view to talk a little bit about the extraordinary role model he provided to me.

Since there is no doubt that Jean-Christophe’s mathematical work will be always a recurring theme in this blog, for now I prefer to postpone all purely mathematical discussions/hommages. Instead, this post contains some `souvenirs’ of my 9 years of friendship with Jean-Christophe as a way to pay a modest tribute for all fantastic and unforgettable moments (both inside and outside Mathematics) shared with him.

**1. Prologue **

I saw Jean-Christophe for the first time during the International Conference on Dynamical Systems held at IMPA in July 2000. Indeed, Jean-Christophe entered the auditorium Ricardo Mañé accompanied by his son (coincidently born in 1984 like me) and wife to deliver a talk about his joint work (of 217 pages) with Jacob Palis (published in 2009).

I remember that I got lost after the first 5 minutes of Jean-Christophe’s lecture, mainly because at that time my mathematical background (of a first year PhD student) was not appropriate.

Remark 1Retrospectively, I find it funny that, even though I could not understand this talk at the time, I would join Jean-Christophe and Jacob in 2009 in the investigation of some open questions left open by their paper.Indeed, I started working with them on this subject by accident: I was present at Collège de France when the offprints of this paper of Jean-Christophe and Jacob arrived by regular mail, and we started right away to “celebrate” it by drinking a coffee while informally discussing the problems left open in this work; at some point, our informal conversation became serious because we noticed an argument allowing to improve the information on the stable sets of the so-called non-uniformly hyperbolic horseshoes.

After this initial success, Jean-Christophe laughed out while saying (as a joke) that we should ask the journal to stop printing the article so that we could add to it an appendix with our brand new result, and, in a more serious tone, he proposed to use this “low-hanging fruit” as the beginning of a separate article (still in prepartion …) on this topic.

**2. My first conversation with Jean-Christophe **

After finishing my PhD in 2004, I started looking for post-doctoral positions. In 2006, Jean-Christophe was visiting IMPA (and also attending ICMP 2006), and my friend Gugu (an IMO gold medalist who finished his PhD at the age of 20 years-old) strongly encouraged me to ask Jean-Christophe whether he could eventually supervise my post-doctoral `sejour’ in France.

Despite Gugu’s advice, I was hesitating to approach Jean-Christophe. In fact, even though a transition from Brazil to France would be relatively smooth under Jean-Christophe’s supervision (partly because his wife was Brazilian and, thus, he spoke Portuguese fluently), I was not sure that I could handle the pressure of becoming the third Brazilian post-doctoral student of a brilliant French mathematician whose expectations about Brazilian post-doctoral students were very high: indeed, his previous experience with Brazilians was to supervise the post-doctoral `sejours’ of Gugu himself and Artur Avila.

After many conversations with Gugu and my wife, I finally got sufficiently motivated to knock the door of Jean-Christophe’s temporary office at IMPA, introduce myself and ask him to be my post-doctoral supersivor: after all, at worst I would simply get a `no’ as an answer.

So, I knocked the door of his office and he called me in (by saying `Oui, entrez’). I told him that I was looking for a post-doctoral advisor, I asked if he could advise me, and, when I was about to give him more details about the kind of Mathematics I have done so far, he gave me a big smile and he said: “There is no need for formal presentations: Gugu talked to me about you and it would be an honor to *me* to be your post-doctoral advisor.”

Such a kind and humble response was certainly not what I was expecting. In particular, I was still under the `shock’ when I told him that I was very happy to hear his answer. Then, I quickly left the office, and my wife (who was waiting for me) saw my puzzled face and asked me: “So, what he said? Do you think the conversation went well?”. Here, I simply replied: “I’m not sure, but I think the conversation went well: if I heard it correctly, he told that it would be an honor to *him* to be my advisor.”

Of course, these two traits (kindness and humbleness) of Jean-Christophe’s personality are well-known among those who met him: in particular, Gugu was not surprised by his answer to me (and this is probably why Gugu insisted that I should talk to Jean-Christophe in the first place!).

**3. My post-doctoral `sejour’ in France **

Several bureaucratic details made that I started my post-doctoral `sejour’ in France in September 17, 2007, i.e., almost one year after my first conversation with Jean-Christophe.

Since the first day of my post-doctoral `sejour’, Jean-Christophe was always very kind to me. For example, he offered me an office next to his own office, so that we could drink coffee, have lunch together and talk (in Portuguese) about many topics (including Mathematics) regularly.

Also, he would never hesitate to stop his research activities to help me with daily problems (e.g., openning a bank account to get my first salary): indeed, his constant support made that my first two years in France were quite smooth and, of course, this gave me the time needed to learn French.

Remark 2Here, he gave the following precious advice: I would learn French more easily by seeing the news on TV because it is easier to absorb the information when just one person is talking calmly at a reasonably constant pace (indeed, this is what he did to learn Portuguese). Moreover, he put me in close contact with the secretaries, so that I would be forced to practice my French (because it was not possible to shift to Portuguese with them).

On the other hand, despite all his attention towards me, he left me *completely free* during the first year of my post-doctoral `sejour’. In fact, as he told me later, he thought that the post-doctoral `sejour’ was an important moment to develop our own ideas about Mathematics and, hence, it was not a good idea to impose to me any specific problem / research topic. In particular, he would limit himself to ask what I was doing recently and periodically invite me to give a talk in the Eliasson-Yoccoz seminar (so that the community would know what I was working on).

Consequently, we would start working in our first joint paper only in 2008…

**4. Our first collaboration **

Upon my return to Paris from a conference in Trieste (in August 2008), I talked to Jean-Christophe about the recently discovered example by Giovanni Forni and myself — later baptized Ornithorynque by Vincent Delecroix and Barak Weiss — of a translation surface with peculiar properties.

He got interested by the subject, and I started to explain to him the main features of the example: “the symmetries of this example are very particular because the Hodge structure on its cohomology …” After letting me end the description of the example, Jean-Christophe replied that, even though my explanation was mathematically correct, he was not happy because it does not allow to `put your hands in the example’. In fact, his phrase (who made a profound impact on me) was: “I don’t like to work on extremely abstract theories with highly sophisticated arguments. I prefer to understand things from a concrete point of view, by working with many concrete examples before reaching the final result. In particular, my `tactics’ is to cover the ground slowly via basic examples before exploring general theories.”

So, he thought that it could be a good idea to work together on a paper giving a explanation for the example in such a way that it could reveal more examples (ideally infinite families) with similar properties.

Frankly speaking, I also found that the original (Hodge-theoretical) arguments obtained with Giovanni had the drawback that they did not allow one to `touch’ the example, and, for this reason, I accepted Jean-Christophe’s offer to investigate more closely the Ornithorynque.

The first meetings related to our work in the Ornithorynque were quite curious: while he would never refuse to meet anytime in the morning (9h, 8h, 7h30, …), we usually would stop our discussions by 3:30 PM or 4 PM because, as he liked to say, by this time “he had runned out of energy”.

At some point, I asked him what time he used to arrive at his office (so that I could try to maximize the span of our conversations). He simply smiled and said: “Normally I wake up around 4:30 or 5 AM, I take the first train and I arrive here around 6 AM.” (Of course, this explains why he could meet me in the morning at any time.) In fact, thanks to his metabolism, Jean-Christophe just needed to sleep about 6 hours per night.

After struggling a bit with my own metabolism, I managed to adapt myself to Jean-Christophe’s rythm and this lead us to a small `competition’ to know how would arrive first at the office. Normally, I would arrive by 6:15 AM at Collège de France and, evidently, Jean-Christophe would be waiting for me with a bottle of fresh coffee (that he had prepared a few minutes ago at the kitchen of Collège), so that we would drink coffee together and talk about the latest news (on a variety of subjects: politics, chess, soccer, etc.) before starting our mathematical conversations. (In general, we would talk from 7 AM to 11:30 AM, take a break to have lunch together, and then come back to work until 3:30 PM or 4 PM.) However, it happened a couple of times that I managed somehow to arrive first in Collège: in every such occasion, Jean-Christophe would spend a couple of minutes explaining why he was `late’. Indeed, this was the natural attitude to him because Jean-Christophe was someone who liked to do his best in everything regardless it was a `small thing’ (e.g., arriving before me at Collège) or a `big thing’ (e.g., proving theorems).

Our work on the `Ornithorynque project’ was going well: the concrete approach of Jean-Christophe (computing tons of particular cases [including multiplying many 4×4 matrices by hand] at an extremely fast pace [which always forced me to be extremely concentrated to be able to follow him up…] for several hours in a row, always keeping an eye for `symmetries’ to reduce the sizes of calculations, etc.) introduced me to a whole new way of doing Mathematics. In particular, there is no doubt that my own vision of Mathematics completely changed by seeing so closely how the mind of a brilliant mathematician works. Moreover, despite the enormous differences in our mathematical skills, Jean-Christophe was always open to hear my ideas and suggestions (and I will always be grateful to him for such an humble attitude towards me).

The rapid and steady progress in `dissecting the Ornithorynque’ made that I did not want to completely stop working on it during the summer vacations in August 2009. So, I had no doubts that I should accept Jean-Christophe’s invitation to spend 10 days in his vacations house in Loctudy (in French Brittany). Since the vacations are sacred in France (it is usually very hard to contact French friends in August …), Jean-Christophe said that the amount of mathematical work in these 10 days would be decided by `chance’, or, more accurately, by the weather. More precisely, we did the following agreement: we would start our day around 5h30 and work until 8h30; if the weather was nice (i.e., not raining too much) outside, we would stop talking about Mathematics and we would go for a walking in the forest or a boat trip in nearby islands, etc.; otherwise, we stay at home working on the Ornithorynque.

In principle, this agreement meant that we would work about of the time because the weather in French Brittany is very unstable: one might see several `seasons’ within a given day …. However, as Jean-Christophe pointed out later, I was extremely lucky that it rained only in three days, so that we had plenty of leisure: we played pétanque, visited Île-de-Sein, etc.

The photos below illustrate such moments (during my second visit [in 2011] to Loctudy): the first picture displays Jean-Christophe plotting a strategy to improve the performance of our participation in a pétanque tournement in Loctudy (but my bad skills in this game made that we were kicked out of the tournement after three matches) and the next two photos show him in Île-de-Sein.

Remark 3In Loctudy’s pétanque club, almost everyone knew that the well-known joke that `you can give mathematicians a hard time by asking them to split a bill in a restaurant’ did not apply to Jean-Christophe. In fact, the `standard procedure’ in the end of the tournement was to go to the bar, ask for a drink and wait for Jean-Christophe to compute (within 2 or 3 minutes) the rankings for the 42 or so participants, sizes of prizes, etc. Also, I overheard a conversation between a newcomer asking an organizer of the tournement: the newcomer was puzzled why everybody went take a drink instead to doing laborious calculations for setting rankings of participants, and the organizer smiled and said: “There is no need to worry. That guy over there will take care of all calculations in 2 minutes, and he knows what he is doing: indeed, he is a sort of `Nobel prize’ in Mathematics…”

**5. Some anecdotes of Jean-Christophe **

Closing this post, I would like to share some anecdotes about Jean-Christophe: in fact, instead of describing more details of our subsequent collaborations, I believe that the reader might get a better idea of Jean-Christophe’s personality via a few anecdotes about other aspects of his life.

Jean-Christophe participated twice in Samba school parades in Rio de Janeiro, and he always laughed at the fact he had all kinds of experiences in these parades: in his first participation, his Samba school won the parade, but in his second participation, his Samba school got the last place…

Jean-Christophe liked to laugh at the fact that if Internet existed before, then the failed attempts of many countries to name the bird `Turkey’ after its country of origin could have been avoided: indeed, the bird is called `Turkey‘ in English, ‘Dinde‘ (a variant of `D’Inde’, i.e., `from India’) in French, `Peru‘ in Portuguese, but the bird is originally Mexican.

Jean-Christophe was a strong chess player: his ELO rating was 2200+ at some point and we followed together many chess tournements via Chessbomb Arena (with particular attention to TCEC, a tournement for chess engines) during some breaks in our mathematical conversations, and he has many interesting anecdotes involving chess.

For example, he wrote here that his PhD advisor (Michel Herman) was so afraid that Jean-Christophe was spending too much time with Chess instead of Mathematics that he called Jean-Christophe’s parents for a conversation about his worries.

Also, once I invited Jean-Christophe, Gugu and Artur Avila for a barbecue at the vacations house of my wife’s family in Vargem Grande, Rio de Janeiro. At some point, we decided to play chess and, as usual, Gugu wanted to play against Jean-Christophe. For some reason, we thought that it could be a good idea to ask Jean-Christophe to play blindfold (with Artur and I moving the pieces in his place), while Gugu would play normally. After 25 moves or so, Gugu announces his move. Jean-Christophe takes a deep breath, spend a couple of minutes thinking, and then asks: `Gugu, are you sure about your move?’. Gugu says `yes’ and Jean-Christophe announces a checkmate in a couple of moves. Gugu is puzzled by Jean-Christophe’s claim, so he looks at the board and he comes up with the genius phrase in response to his blindfold friend: `Is it already checkmate? *I could not see it*!’ Of course, Jean-Christophe was right and, after laughing for a couple of minutes at Gugu’s phrase, we passed to the next match (Jean-Christophe versus me…).

Jean-Christophe liked sports in general: during his youth, he played rugby and he sailed together with his friends from École Normale Supérieure, and he followed on TV and newspapers many tournements (soccer, rugby, golf, snooker, …). In particular, we saw together several finals of the UEFA Champions League while drinking his favorite beer (namely, Guinness) and he followed on the hospital’s TV the Rio 2016 Olympic Games (which was especially interesting to him because he knew `Rio’s Zona Sul’ [where rowing and cycling took place]).

Finally, Jean-Christophe was an avid reader and an erudite person: he read (in French, English and Portuguese) many books and novels per month, he saw many movies, and he liked to go to museums. For example:

- he was happy when he learned that I read The Murders in the Rue Morgue (partly because Edgar Allan Poe was one of his favorite writers),
- he borrowed me Crimes Cèlebres by Alexandre Dumas and Ne le dis à personne (French version of a thriller novel) by Harlan Corben, so that I could improve my French,
- besides the classical in Brazilian literature (by Machado de Assis, José de Alencar, etc.), Jean-Christophe liked `Agosto’ by Rubem Fonseca. In fact, Jean-Christophe told me that he read the original book in Portuguese because he was not sure that the French translation was very good: indeed, the title of `Agosto’ — literally `August’ in English — was translated as `Un été brésilien‘ — `a Brazilian summer’ — reveals that the editors probably did a job of questionable quality since it is winter (and not summer) in Brazil in August …
- he was a big fan of Alfred Hitchock (among many other film directors): in particular, I will always remember with affection a `Hitchcock session’ in Jean-Christophe’s vacations house in Loctudy.
- after trying to read (unsuccessfully) `À la recherche du temps perdu‘ during his youth, Jean-Christophe spent the last months of his life (doing Mathematics and) re-reading all seven volumes of Proust’s novel …

**6. Epilogue**

Since a picture is worth more than a thousand words, I’m sharing below a couple of photos of Jean-Christophe (holding my daughter, next to my wife [during vacations in French Brittany], working with me at Mittag-Leffler institute on our joint paper with Martin Moeller, and together with Gugu, Artur, Fernando Codá Marques, Susan Schommer and myself in Artur’s appartment, respectively).

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As it turns out, Yoccoz’s proof of (1) is indirect: first, he introduces the notion of *strongly regular parameter* and he proves that strongly regular parameters are a special case of regular parameters; secondly, he exploits the nice features of strong regularity to *transfer* some key properties from the phase space to the parameter space in order to prove that

Today, we shall define strong regularity and prove the regularity of such parameters (while leaving the proof of (2) for the final post of this series).

**1. Some preliminaries **

** 1.1. Quick review of the regularity property **

For , has two fixed points and with . Note that the critical value belongs to .

In a certain sense, the key idea is to study the dynamics of via certain intervals (“Yoccoz puzzle pieces”) bounded by points in the pre-orbits of .

For example, the notion of regular parameter was defined with the aid of the intervals and where is given by . Indeed, is *regular* if there are and such that

for all . Here, is called –*regular* if there are and an interval such that sends diffeomorphically onto in such a way that . For later use, we denote by the inverse branch of restricted to .

In general, any -regular point belongs to a *regular interval* of order , that is, an interval possessing an open neighborhood such that sends diffeomorphically onto in such a way that . In other words, the set of -regular points is the union of regular intervals of orders .

It is easy to describe regular intervals (“Yoccoz puzzle pieces”) in terms of the pre-orbits of . In fact, denote by (so that and ). It is not difficult to check that if is a regular interval of order and is the associated neighborhood, then are *consecutive* points of and are *consecutive* points of .

** 1.2. Dynamically meaningful partition of the parameter space **

For later use, we organize the parameter space as follows. For each , we consider a maximal open interval such that is the first return of to under for all .

In analytical terms, we can describe the sequence as follows. For , let be and, for , define recursively as

In these terms, is the solution of the equation .

Remark 1By definition, . This inductive relation can be exploited to give that for all and .From this analytical definition of , one can show inductively that for along the following lines.

This estimate allows us to show that the function has derivative between and for . Since this function takes a negative value at and a positive value at , we see that this function has a unique simple zero such that for , as desired.

Remark 2Note that is a decreasing sequence such that for some universal constant . Indeed, the function takes the value at (cf. Subsection 4.2 of the previous post), it vanishes at , and it has derivative between and , so that .

From now on, we think of where is a *large* integer.

**2. Strong regularity **

Given , let be the collection of maximal regular intervals of positive order contained in and consider

the function for , and the map ( for ): cf. Subsection 4.3.3 of this post here.

Remark 3Even though is not contained in any element of , we set and for .

The elements of of “small” orders are not hard to determine. Given , define by:

It is not difficult to check that the *sole* elements of of order are the intervals

and, furthermore, any other element of has order .

The intervals , , are called *simple regular intervals*: this terminology reflects the fact that they are the most “basic” type of regular intervals.

In this setting, a parameter is *strongly regular* if “most” of the returns of to occur on simple regular intervals:

Definition 1We say that is strongly regular up to level if and, for each , one has

A parameter is called strongly regular if it is strongly regular of all levels .

Remark 4Let be a strongly regular parameter. It takes a while before encounters a non-simple regular interval: if (or, equivalently, ), then (3) implies that

where . In particular, , so that the first iterates of encounter exclusively at simple regular intervals.

**3. Regularity of strongly regular parameters **

Let us now outline the proof of the fact that strongly regular parameters are regular.

** 3.1. Singular intervals **

Given , we say that an interval is –*singular* if its boundary consists of two consecutive points of , but is not contained in a regular interval of order . The collection of -singular intervals is denoted by .

Remark 5For and , there is only one -singular interval, namely . For and , there are exactly three -singular intervals, namely and .

By definition, .

For later reference, denote . In these terms, is regular whenever there are such that

As a “warm-up”, let us show the following elementary fact:

for all and .

*Proof:* For , we have that is a singleton (cf. Remark 5). Moreover,

when (cf. Subsection 4.2 of the previous post). Since the function is increasing on , we get the desired estimate for .

On the other hand, if , then

This completes the proof of the proposition.

** 3.2. Central, peripheral and lateral intervals **

The analysis of for requires the introduction of certain (combinatorially defined) neighborhoods of the critical point and the critical value .

Assume that for some . For each , let be the element such that .

Denote by the decreasing sequence of regular intervals containing the critical value defined recursively as follows: is a regular interval of order and is the regular interval of order determined by its inverse branch

Also, let us consider and . Here, if is a regular interval, then , where and , , , and in general.

Remark 6By definition, the endpoints of are the points of immediately adjacent to the endpoints of .

Note that is the connected component of containing the critical point , while the endpoints of are adjacent in to the endpoints of . Here (and in the sequel), .

We say that an interval is *central*, *lateral* or *peripheral* depending on its relative position with respect to :

Definition 3Let be a strongly regular parameter up to a level such that . An interval is called:

- central whenever ;
- lateral if but ;
- peripheral if .

** 3.3. Measure estimate for central intervals **

We shall control the total measure of central intervals by estimating the Lebesgue measure of :

Proposition 4Let be a strongly regular parameter up to level . Then,

for all .

*Proof:* is the neighborhood of the critical point of the quadratic map defined by . Therefore, is comparable to :

By the usual distortion estimates (cf. Subsection 4.3 of the previous post), it is possible to check that is comparable to :

This reduces our task to estimate . Since the interval has a fixed size and for some (as ), it suffices to control for . For this sake, we recall that the derivative of is not far from a “coboundary”:

where (see Proposition 6 of the previous post for a motivation of in the case ). In particular,

By exploiting this estimate, one can show (with a one-page long argument) that the strong regularity up to level of implies a (strong form of) *Collet-Eckmann condition*:

for all and . Because for and for , the proof of the proposition is complete.

** 3.4. Measure estimates for peripheral intervals **

We control the total measure of peripheral intervals by relating them to singular intervals of lower order. More concretely, a (half-page long) *combinatorial* argument provides the following *structure* result for the generation of peripheral intervals:

Proposition 6 (Structure of peripheral intervals)Let be strongly regular up to level and consider . If is a peripheral interval, then:

- either has the form for some ,
- or has the form for some ,

where (is a regular interval of order ).

Corollary 7Let be strongly regular up to level and fix . Then, the total measure of peripheral -singular intervals is

*Proof:* A point in a peripheral interval is not close to : indeed, (by definition) and (by Corollary 5 for ). Hence,

The previous proposition says that if is a peripheral interval, then has the form with or with . From the fact that the derivative of is an “almost coboundary” (cf. the proof of Proposition 4 above), one can show that:

- when ;
- when .

Therefore, for some or and, *a fortiori*,

for some or .

It follows that

so that the proof of the corollary is complete.

** 3.5. Measure estimates for lateral intervals **

The anlysis of lateral intervals is combinatorially more involved. For this reason, we subdivide the class of lateral intervals into *stationary* and *non-stationary*:

Definition 8The level of is the largest integer such that . (Note that and .Let be strongly regular up to level , fix and consider a lateral interval. For each , either or (because ).

We say that the level is stationary if .

The strategy to control the total measure of lateral intervals is similar to argument used for peripheral intervals: we want to exploit structure results describing the construction of lateral intervals out of singular intervals of lower orders. As it turns out, the case of lateral intervals with stationary level is somewhat easier (from the combinatorial point of view) and, for this reason, we start by treating this case.

For later use, we denote

**3.5.1 Lateral intervals with stationary levels**

Denote by (a regular interval of order ) and . The structure of lateral intervals with stationary levels is given by the following proposition:

Proposition 9Let be a lateral interval with stationary level . Suppose that reverses the orientation. Then, has the form for some contained in .A similar statement (with replacing ) holds when preserves orientation.

The proof of this proposition is short, but we omit it for the sake of discussing the total measure of lateral intervals with stationary level.

*Proof:* The argument is very close to the case of peripheral intervals (i.e., Corollary 7). In fact, if is a lateral interval with stationary level and reverses the orientation, then for some with . Note that implies that . Since , the usual bounded distortion properties say that

for any and, *a fortiori*,

This completes the proof of the corollary because (cf. the proof of Proposition 4).

**3.5.2 Lateral intervals with non-stationary levels**

The structure of lateral intervals with non-stationary levels is the following:

Proposition 11Let be a lateral interval with non-stationary level . Suppose that reverses the orientation. Then,

- either for some ,
- or where is a regular interval of order (with right endpoint immediately to the left of in ) and .

A similar statament holds when preserves orientation.

By exploiting this structure result in a similar way to the arguments in Corollaries 7 and 10, one can show (with a half-page long proof) the following estimate:

Corollary 12Assume that is a non-stationary level. Then,

Moreover, if , then one has a better estimate:

** 3.6. Proof of regularity of strongly regular parameters **

The measure estimates developed in the last three subsections permit to establish the main result of this post, namely:

Theorem 13Fix . If is large enough depending on (i.e., ), then for any strongly regular up to level and any we have

In particular, any strongly regular parameter is regular.

*Proof:* We will show this theorem by induction. The initial cases of this theorem were already established in Proposition 2.

Suppose that and for all . Replacing by a smaller integer (if necessary), we can assume that .

By definition of central, lateral and peripheral intervals,

By Corollary 5, the first term satisfies:

The right hand side is at most whenever

i.e.,

We have two possibilities:

- if , then for large enough (e.g., );
- if , we have for large enough thanks to the strong regularity assumption (cf. Remark 4).

The contribution of peripheral intervals is controlled by induction hypothesis. More precisely, by Corollary 7, one has

By induction hypothesis, we conclude that

The contribution of lateral intervals is estimated as follows. Fix . The bounds on in the case of a non-stationary level are worse than in the case of a stationary level : compare Corollaries 10 and 12. For this reason, we will use only the bounds coming from the non-stationary situation in the sequel.

If is not simple, i.e., its order is , then the induction hypothesis (applied to ) implies that

Since for large enough and (cf. Remark 4), we conclude that

for large enough.

If is simple (i.e., ), then we use the second part of Corollary 12 and the induction hypothesis to obtain:

Thus,

for large enough.

The last two inequalities together imply that

Finally, by plugging the estimates (5), (6) and (7) into (4), we deduce that

for large enough. This proves the desired theorem.

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An interesting corollary of our results is a refinement of a theorem of Chernov on the number of periodic points of certain billiard maps. More precisely, for a certain class of billiard maps , Chernov proved that

where is the set of periodic points of with period and is the Kolmogorov-Sinai entropy of the Liouville measure. From our main results, Yuri and I can show that the billiard maps studied by Chernov actually satisfy:

Remark 1Our improvement of Chernov’s theorem is “similar in spirit” to Sarig’s improvement of Katok’s theorem on the number of periodic points for smooth surface diffeomorphisms: see Theorem 1.1 in Sarig’s paper for more details.

Below the fold, we give a slightly simplified version of the main result in our paper and we explain some steps of its proof.

**1. Symbolic models for certain billiard maps **

Consider a planar billiard map , , where is a compact billiard table whose boundary is a finite union of smooth curves: by definition, whenever the straight line starting from in direction hits at with angle of incidence ( angle of reflection) .

Recall that a billiard map preserves the Liouville measure .

In 1986, Katok and Strelcyn showed that the so-called Pesin theory of smooth non-uniformly hyperbolic diffeomorphisms could be extended to non-uniformly hyperbolic billiard maps under mild conditions.

More concretely, a billiard map usually exhibits a *singular set* (related to discontinuities of , grazing collisions, etc.) and, roughly speaking, Katok and Strelcyn results say that if has reasonable geometry (e.g., the Liouville -measure of -neighborhoods of decay polynomially fast with ), then Pesin theory applies to a non-uniformly hyperbolic billiard map whose first two derivative explode *at most polynomially fast* as one approaches .

The class of billiard maps within the range of Katok-Strelcyn theory is vast: it includes Sinai billiards, Bunimovich stadia and asymmetric lemon billiards.

Philosophically speaking, the basic idea behind Katok-Strelcyn theorems is that the good exponential behavior provided by non-uniform hyperbolicity is strong enough to overcome the bad polynomial behavior near the singular set . (Of course, this is easier said than done: Katok-Strelcyn’s work is extremely technical at some places.)

In our paper, Yuri and I show that Katok-Strelcyn philosophy can also be used to extend Sarig’s theory to billiard maps:

Theorem 1Let be any billiard map within the framework of Katok-Strelcyn’s theory (e.g., Sinai billiards, Bunimovich stadia, etc.). Then, there exists a topological Markov shift (of countable type) and a Hölder continuous map such that

- the shift codes the dynamics of , i.e., ;
- most -orbits are captured by the coding, i.e., the set has full Liouville -measure;
- is finite-to-one (and, hence, the Liouville measure on can be lifted to without increasing the entropy).

Remark 2The main result of our paper (Theorem 1.3) deals with a more general class of surface maps with discontinuities, but its precise statement is somwhat technical: we refer the curious reader to the original article for more details.

**2. Sarig’s theory of symbolic models **

The general strategy to prove Theorem 1 follows closely Sarig’s methods. More precisely, given a billiard map such as a Sinai or Bunimovich billiard, we fix such that the Lyapunov exponents of with respect to the Liouville measure do not belong to the interval .

By Oseledets theorem, there is a set of full -measure such that any has the following properties:

- for all , there are unit vectors with for ;
- and ;
- the angle between and decays subexponentially:

Furthermore, the assumption that the singular set has a reasonable geometry (e.g., the logarithm of the distance to is -integrable) says that the subset consisting of points whose -orbits do *not* approach exponentially fast, i.e.,

also has full -measure.

One of the basic strategies to code (a full measure subset of) relies on the so-called shadowing lemma: very roughly speaking, for sufficiently small, we want the -orbit of -almost every to be shadowed by (“fellow travel with”) finitely many –*generalized pseudo-orbits*.

The notion of -generalized pseudo-orbits is the same from Sarig’s work: in particular, they are *not* defined in terms of sequences of points chosen from a countable dense subset with the property that for all , but rather in terms of sequences of *double Pesin charts* (taken from a countable “dense” subset ) with the property that –*overlaps* for all .

Here, the advantage in replacing points by Pesin charts comes from the fact that looks like a uniformly hyperbolic linear map, so that we can hope to apply the usual tools from the theory of hyperbolic systems (stable manifolds, etc.) to establish the desired shadowing lemma.

After this succint explanation of Sarig’s method for the construction of symbolic models for non-uniformly hyperbolic systems, let us now discuss in more details the implementation of Sarig’s ideas.

** 2.1. Linear Pesin theory **

Before trying to render into an almost linear hyperbolic map in adequate (Pesin) charts, let us convert the derivative of at into a uniformly hyperbolic matrix. For this sake, we use an old trick in Dynamical Systems, namely, we introduce the hyperbolicity parameters

and angle between and . Note that and are well-defined (i.e., the corresponding series are convergent) because .

In terms of these parameters, we can define the linear map via

where is the canonical basis of .

A straightforward computation reveals that becomes a uniformly hyperbolic matrix when viewed through the linear maps , i.e.,

where and .

Of course, the conversion of the non-uniformly hyperbolic map into a uniformly hyperbolic matrix has a price: while the norm of is , a simple calculation shows that the Frobenius norm of its inverse is

In particular, “explodes” when the hyperbolicity parameters degenerate (e.g., approaches zero).

** 2.2. Non-linear Pesin theory **

After converting into a uniformly hyperbolic matrix via , we want to convert into an almost (uniformly hyperbolic) linear map near . For this sake, we compose with the exponential map to obtain the *Pesin chart*

In this way, is a map fixing such that

where and .

Of course, this means that is an almost (hyperbolic) linear map in some neighborhood of , but this qualitative information is not useful: we need to control the size of this neighborhood of (in order to ensure that a *countable* set of [double] Pesin charts suffice to code the dynamics of on a full -measure set of points of ).

In this direction, we introduce a small parameter depending on , and the distance of to (whose precise definition can be found at page 10 in our paper). Then, a simple calculation (cf. Theorem 3.3 in our paper) shows that, for all in the square , one has

where and are smooth functions whose -norms on are smaller than .

In fact, our choice of involves and in order to control the distortion create by the linear maps and in the definition of .

On the other hand, the dependence of on is a novelty with respect to Sarig’s paper and it serves to control the eventual polynomial explosion of the first two derivatives and of near (i.e., and for some ).

Once we dispose of good formulas for on the Pesin charts of and , we want to “discretize” the set of Pesin charts: since our final goal is to code most -orbits with a *countable* set of Pesin charts, we do not want to keep all , .

Here, the basic idea is that we can safely replace by whenever has (essentially) the same features of , i.e., it is an almost (hyperbolic) linear map on the square .

Since is defined in terms of , it is not surprising that and and, *a fortiori*, and are close whenever the points and are close *and* the matrices and are close.

This motivates the definition of -overlap of two Pesin charts.

Definition 2Given and , denote by the restriction of to the square . We say that -overlaps if and

As the reader might suspect, this definition is designed so that if -overlaps , then the hyperbolicity parameters , , of and are close, and is –-close to the identity (on a square for some ): see Proposition 3.4 in our paper.

By exploiting this information, we show (in Theorem 3.5 of our paper) that if -overlaps , then

where and are smooth functions whose -norms on are smaller than .

** 2.3. Generalized pseudo-orbits **

The graph associated to the topological Markov shift coding will be defined in terms of two pieces of data: its vertices are –*double charts* and its edges connect a double chart whose “iterate” under has -overlap with another double chart.

Definition 3A -double chart is a pair of Pesin charts whose parameters belong to the countable set .

Remark 3The philosophy in the consideration of and is that, contrary to the uniformly hyperbolic case, the forward and backward behavior of non-uniformly hyperbolic systems might be very different, hence we need to control them separately.

Definition 4Given -double charts and , we draw an edge whenever

- (GPO1) -overlaps and -overlaps .
- (GPO2) and .

Remark 4GPO stands for “generalized pseudo-orbit”. The second condition (GPO2) is a greedy way of ensuring that the parameters and (controlling , and, thus, the hyperbolicity parameters , ) are the largest possible.

Definition 5A -generalized pseudo-orbit is a sequence of -double charts such that we have an edge for all .

The fact that -generalized pseudo-orbits are useful for our purposes is explained by the following result (cf. Lemma 4.6 in our paper):

Lemma 6Every -generalized pseudo-orbit shadows an unique point, i.e., there exists an unique such that

for all .

The proof of this *shadowing lemma* follows the usual ideas in Dynamical Systems: first, one defines stable/unstable manifolds using the Hadamard-Perron graph transform method, and, secondly, one shows that the unique point shadowed by is precisely the unique intersection point between the stable and unstable manifolds. In particular, we use here that the fast (exponential) pace of the dynamics along “almost stable/unstable manifolds” (called -admissible manifolds) is sufficiently strong to apply Sarig’s arguments even if and are allowed to explode at a slow (polynomial) pace near the singular set .

** 2.4. Coarse graining **

The next step is to select a *countable* collection of -double charts such that the corresponding -generalized pseudo-orbits shadow a set of full -measure.

Theorem 7For sufficiently small, there exists a countable collection of -double charts such that

- is discrete: for all , the set is finite;
- is sufficient to code most -orbits: there exists of full -measure so that if , then there exists a -generalized pseudo-orbit shadowing ;
- all elements of are relevant for the coding: given , there exists a -generalized pseudo-orbit with that shadows a point in .

In a nutshell, the proof of this theorem is a pre-compactness argument. More precisely, for each in an appropriate subset (of full -measure), we consider the parameters

controlling the Pesin charts , , . Since the spaces , and are pre-compact (or, more precisely, for all , the sets , and are compact), we can select a countable subset of which is *dense* in the following sense: for all and , there exists such that

and, for each ,

In terms of , the countable collection of -double charts verifying the conclusions of the theorem is essentially .

This theorem yields a topological Markov shift associated to the graph whose set of vertices is and whose edges are (cf. Definition 4), i.e., is the set of bi-infinite (-indexed) paths on and is the shift dynamics on . Since any is a -generalized pseudo-orbit, we have a map , where

is the point shadowed by .

The map has the following properties (cf. Proposition 5.3 in our paper):

Proposition 8Every has finite valency in (and, hence, is locally compact). Moreover, is Hölder continuous, and codes most -orbits (i.e., and has full -measure).

The first part of this proposition follows from the discreteness of (cf. Theorem 7), the Hölder continuity of is a consequence of the nice dynamical properties of almost stable/unstable manifolds, and the fact that codes most -orbits (i.e., has full Liouville measure) is deduced from the second item of Theorem 7.

** 2.5. Inverse theorem **

In general, is *not* finite-to-one, i.e., might not satisfy the last conclusion of Theorem 1. Therefore, we need to *refine* before trying to use to induce a locally finite cover of a subset of of full -measure.

For this sake, it is desirable to understand how loses injectivity, and, as it turns out, this is the content of the so-called *inverse theorem* (cf. Theorem 6.1 in our paper):

Theorem 9If are -recurrent and

then all relevant parameters (distance, angle, hyperbolicity, etc.) are close together:

- for all ;
- and for all ;
- for all ;
- for all ;
- for all ;
- for all , is –-close to (for an adequate choice of ) on the square .

Intuitively, this theorem says that “tends” to be finite-to-one because the parameters of (a -recurrent) “essentially” determine the parameters of any (-recurrent) with , so that the discreteness of (cf. Theorem 7) implies that there are not many choices for such .

The proof of the inverse theorem is the core part of both Sarig’s paper and our work. Unfortunately, the explanation of its proof is beyond the scope of this post (because it is extremely technical), and we will content ourselves in pointing out that the presence of the singular set introduces extra difficulties when trying to run Sarig’s arguments: for example, contrary to Sarig’s case, the parameter also depends on , so that we need to take extra care in the discussion of the fourth item of the inverse theorem above.

** 2.6. Bowen-Sinai refinement method **

Once we dispose of the inverse theorem in our toolkit, the so-called *Bowen-Sinai refinement method* (for the construction of Markov partitions) explained in Sections 11 and 12 of Sarig’s paper can be used in our context of billiard maps *without any extra difficulty*: see Section 7 of our paper for more details.

For the sake of convenience of the reader, let us briefly recall how Bowen-Sinai method works to convert the coding into the desired coding satisfying the conclusions of Theorem 1.

First, we start with the collection , where

The s/u-*fiber* of is

where is *any* -recurrent element of with . (The nice properties of “almost stable/unstable manifolds” ensure that is well-defined [i.e., it doesn’t depend on the particular choice of ].)

It is not difficult to show that is a cover of a full -measure subset which is *locally finite* (i.e., for all , the set is finite). Moreover, has *local product structure* (i.e., for all , , ), and is a *Markov cover* (i.e., for any -recurrent with , one has and ).

Now, we refine according to the ideas of Bowen and Sinai. More concretely, we take the Markov cover and, for any , we consider:

Then, we define as the partition induced by the collection

At this point, we can complete the proof of Theorem 1 by proving that is a countable Markov partition such that the graph with set of vertices and edges whenever induces a topological Markov shift

with the desired properties, namely, is Hölder continuous, finite-to-one, and .

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This is the first installment of a series of two articles on the Kontsevich-Zorich cocycle over certain -invariant loci in moduli spaces of translation surfaces obtained from cyclic cover constructions (inspired from the works of Veech and McMullen).

More precisely, the second paper of this series (still in preparation) studies the Kontsevich-Zorich cocycle over -orbits of certain cyclic covers of translation surfaces in hyperelliptic components called and in the literature. (The curious reader can find more explanations about this forthcoming paper in this old blog post here [cf. Remark 7].)

Of course, before studying the cyclic covers, we need to obtain some good description of the Kontsevich-Zorich cocycle on the hyperelliptic components and this is the purpose of the first article of the series.

Since the first paper of this series is not long, this post will just give a brief “reader’s guide” rather than entering into the technical details.

Remark 1In the sequel, we will assume some familiarity with translation surfaces.

**1. Rauzy-Veech groups and Zorich conjecture **

The starting point of our article is a description of certain combinatorial objects — called *hyperelliptic Rauzy diagrams* — coding the dynamics of the Kontsevich-Zorich cocycle on and .

Remark 2By the time that the first version of our article was written, we thought that we found a new description of these diagrams, but Pascal Hubert kindly pointed out to us that G. Rauzy was aware of it.

One important feature of this description of hyperelliptic Rauzy diagrams is the fact that we can order these diagrams by “complexity” in such a way that two consecutive diagrams can be related to each other by an *inductive* procedure.

The behavior of the Kontsevich-Zorich cocycle (with respect to Masur-Veech measures) is described in general by the *Rauzy-Veech algorithm*: roughly speaking, this algorithm is a natural way to attach matrices (acting on homology groups) to the arrows of Rauzy diagrams, and, in this language, the action of the Kontsevich-Zorich cocycle is just the multiplication of the matrices attached to concatenations of arrows of Rauzy diagrams.

In particular, the features of the Kontsevich-Zorich cocycle (with respect to Masur-Veech measures) can be derived from the study of the so-called *Rauzy-Veech groups*, i.e., the groups generated by the matrices attached to the arrows of a given Rauzy diagram. For example, the celebrated paper of Avila and Viana solving affirmatively a conjecture of Kontsevich and Zorich proves the simplicity of the Lyapunov exponents of the Kontsevich-Zorich cocycle by establishing (inductively) the *pinching* and *twisting* properties for Rauzy-Veech groups.

In our article, we exploit the “inductive” description of hyperelliptic Rauzy diagrams to compute the *hyperelliptic* Rauzy-Veech groups.

Remark 3Our arguments for the computation of hyperelliptic Rauzy-Veech groups were inspired from the calculations of -blocks of the Kontsevich-Zorich cocycle over cyclic covers in the second paper of this series. In other words, we first developed some parts of the second paper before writing the first paper.

An interesting corollary of this computation is the fact that hyperelliptic Rauzy-Veech groups are *explicit* finite-index subgroups of the symplectic groups , so that they are *Zariski dense* in .

The Zariski density of (general) Rauzy-Veech groups in symplectic groups was conjectured by Zorich (see, e.g., Remark 6.12 in Avila-Viana paper) as a step towards the Kontsevich-Zorich conjecture established by Avila-Viana. Therefore, the previous paragraph means that Zorich conjecture is true for hyperelliptic Rauzy-Veech groups (and this justifies our choice for the title of our paper).

Here, it is worth to point out that Zorich conjecture asks *more* than what is needed to prove the Kontsevich-Zorich conjecture. Indeed, the Zariski denseness in symplectic groups imply the pinching and twisting properties of Avila-Viana, so that Zorich conjecture *implies* Kontsevich-Zorich conjecture. On the other hand, we saw in this previous blog post that a pinching and twisting group of symplectic matrices might not be Zariski dense: in other words, the techniques of Avila-Viana solve the Kontsevich-Zorich *without* addressing Zorich conjecture.

Thus, our proof of Zorich conjecture for hyperelliptic Rauzy-Veech groups gives an alternative proof of this particular case of Avila-Viana theorem.

**2. Braid groups and A’Campo theorem **

After a preliminary version of our article was complete, Martin Möller noticed some similarities between our characterization of hyperelliptic Rauzy-Veech groups and a result of A’Campo on the images of certain monodromy representations associated to hyperelliptic Riemann surfaces.

As it turns out, this is not a coincidence: we showed that the elements of the hyperelliptic Rauzy-Veech group associated to certain *elementary* loops on hyperelliptic Rauzy diagrams are induced by Dehn twists lifting the generators of a braid group; hence, this permits to recover our description of hyperelliptic Rauzy-Veech groups from A’Campo theorem.

Remark 4In a certain sense, the previous paragraph is a sort of sanity test: the same groups found by A’Campo were rediscovered by us using different methods.

Closing this short post, let me point out that this relationship between loops in hyperelliptic Rauzy diagrams and Dehn twists in hyperelliptic surfaces reveals an interesting fact: the fundamental groups of the combinatorial model (hyperelliptic Rauzy diagram) coincides with the orbifold fundamental groups of and . In other words, the hyperelliptic Rauzy diagrams “see” the topology of objects ( and ) coded by them.

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More recently, he gave a couple of lectures at Collège de France aiming to generalize his proof of Jakobson theorem to obtain some results improving upon the Wang-Young theory of rank one attractors.

Of course, his lectures (whose videos are available at his website) started by recalling several elements of his proof of Jakobson theorem and I decided to take the opportunity to write a series of posts about Jean-Christophe’s proof of Jakobson theorem (based on his hand-written lecture notes from 1997).

For the first installment of this series, we’ll review some aspects of the dynamics of the one-dimensional quadratic maps and, after that, we’ll state Jakobson’s theorem, describe the three main steps of Yoccoz proof of Jakobson theorem and implement the first step of the strategy.

**1. Introduction **

The dynamics of one-dimensional affine maps is fairly easy understand. The change of variables provided by the translation transforms into . If , the choice shows that the affine map is conjugated to its linear part . If , is a translation.

In other words, the dynamics of polynomial maps is not very interesting when . On the other hand, we shall see below that quadratic maps are already sufficiently complicated to produce all kinds of rich dynamical behaviors.

**2. Quadratic family **

The *quadratic family* is where and .

Remark 1This family is sometimes presented in the literature as or . As it turns out, these presentations are equivalent to each other: indeed, the affine change of variables converts into

The dynamics of near infinity is easy to understand: in fact, since for , one has that attracts the orbit of any with .

This means that the interesting dynamics of occurs in the filled-in Julia set:

Note that is totally invariant, that is, . Also, is a compact set because implies that and whenever , so that

(where ).

Moreover, because it contains all periodic points of (i.e., all solutions of the algebraic equations , ).

Remark 2is a full compact set, i.e., is connected: indeed, this happens because the maximum principle implies that a bounded open set with boundary must be completely contained in (i.e., ).

The dynamics of on is influenced by the behaviour of the orbit of the critical point . More precisely, let us consider the Mandelbrot set

Indeed, we shall see in a moment that there is a substantial difference between the dynamics of on depending on whether or .

Remark 3If , then one can check by induction that for all . Thus, the Mandelbrot set is the compact set

Furthermore, the Mandelbrot set is symmetric with respect to the real axis: if and only if .Also, the maximum principle implies that is full (i.e., is connected).

Moreover, intersects the real axis at the interval : indeed, we already know that ; on the other hand, the critical orbit is trapped between the fixed point (solving the equation ) and its preimage under for , so that ; finally, the fixed point moves away from the real axis for and this allows for the critical orbit to escape to infinity, so that .

Let us analyze the dynamics of on . For this sake, let us take a sufficiently large open disk centered at the origin (and containing the critical value ). The preimage is a topological open disk whose closure is contained in , and is a degree two covering map.

By induction, we can define inductively . We have two possibilities:

- (a) if the critical value belongs to , then is a topological disk whose closure is contained in , and is a degree two covering map;
- (b) if for some , then is the union of two topological disks and whose closures are disjoint, and and are univalent maps with images ; in particular, by Schwarz lemma, , , are (uniform) contractions with respect to the Poincaré (hyperbolic) metrics on and , the filled-in Julia set
is a Cantor set, and the dynamics of on is conjugated to a full unilateral two-shift , , via , where .

Remark 4Observe that, in item (b) above, we got analytical information ( is an uniform contraction) from topological information ( is an open topological disk with ) thanks to some tools from Complex Analysis (namely, Schwarz lemma). This type of argument is a recurrent theme in one-dimensional dynamics.

Note that the situation in item (b) occurs if and only if the critical orbit escapes to infinity, i.e., . Otherwise, the sequence of nested topological disks are defined for all and .

In summary, the dynamics of on falls into the classical realm of hyperbolic dynamical systems when (cf. item (b)).

From now on, we shall focus on the discussion of for . Actually, we will talk exclusively about *real* parameters in what follows.

**3. Statement of Jakobson’s theorem **

Suppose that has an *attracting* periodic orbit of period , i.e., and .

It is possible to show that the *basin of attraction*

must contain the critical point . Thus, given , there exists at most *one* attracting periodic orbit of .

Remark 5If has an attracting periodic orbit , then is hyperbolic in . Indeed, this follows from the same argument in item (b) above: the basin of attraction is the interior of and the boundary (called Julia set in the literature) is disjoint from the closure of the critical orbit ; thus, is contained in the open connected set with , so that the univalent map from to a component of is a local contraction with respect to the Poincaré metric on ; since is a compact subset of , it follows that there exists such that for all and , i.e., is uniformly expanding on .

Denote by the set of parameters such that has an attracting periodic point. The hyperbolicity of on for permits to show that is contained in the *interior* of the Mandelbrot set .

The so-called *hyperbolicity conjecture* of Douady and Hubbard in 1982 asserts that coincides with the interior of . This conjecture is one of the main open problems in this subject (despite partial important progress in its direction).

Nevertheless, for *real* parameters , we have a better understanding of the dynamics of for *most* choices of .

More precisely, it was conjectured by Fatou that is dense in . This conjecture was independently established by Lyubich and Graczyk-Swiatek in 1997: they showed that hyperbolicity of is *open and dense* property in real parameter space . In particular, this gives an extremely satisfactory description of the dynamics of for a *topologically* large set of parameters .

Of course, it is natural to ask for a *measure-theoretical* counterpart of this result: can we describe the dynamics of for *Lebesgue almost every* ?

It might be tempting to conjecture that has full Lebesgue measure on . Jakobson showed already in 1981 that this naive conjecture is false: does *not* have full Lebesgue measure on .

Theorem 1 (Jakobson)Let be the subset of parameters such that

- there exists such that for Lebesgue almost every (where );
- preserves a probability measure which is absolutely continuous with respect to the Lebesgue measure on .

Then, the Lebesgue measure of is positive and, in fact, has density one near :

(where stands for the Lebesgue measure).

Nevertheless, Lyubich showed in 2002 that has full Lebesgue measure on , so that we get a satisfactory description of the dynamics of for Lebesgue almost every parameters .

**4. Overview of Yoccoz proof of Jakobson theorem **

In 1997 and 1998, Jean-Christophe Yoccoz gave a series of lectures at Collège de France where he explained a proof of Jakobson theorem based on some combinatorial objects for the study of called *Yoccoz puzzles* (see, e.g., the first section of this paper here by Milnor).

Our goal in this series of posts is to describe Yoccoz’s proof of Jakobson theorem by following Yoccoz’s lecture notes (distributed by him to those who asked for them) recently submitted for publication (after this announcement here).

Very roughly speaking, Yoccoz proof of Jakobson’s theorem has three steps. First, one introduces the notion of *regular parameters* and one shows that satisfies the conclusion of Jakobson theorem whenever is a regular parameter. Unfortunately, it is not so easy to estimate the Lebesgue measure of the set of regular parameters. For this reason, one introduces a notion of *strongly regular parameters* and one proves that strongly regular parameters are regular (thus justifying the terminology). Finally, one transfers the estimates on “Yoccoz puzzles” in the phase space to the parameter space (via certain analogs of Yoccoz puzzles in parameter space called “Yoccoz parapuzzles”): by studying how the real traces of Yoccoz puzzles move with the parameter , one completes the proof of Jakobson theorem by showing that

In the remainder of this post, we will implement the first step of this strategy (i.e., the introduction of regular parameters and the derivation of the conclusion of Jakobson theorem for regular parameters via arguments from Thermodynamical Formalism).

** 4.1. Preliminaries **

has two real fixed points and for . Note that is repulsive for all , while is attractive, resp. repulsive for , resp. . Moreover, the real filled-in Julia set equals for all .

The dynamical features of are determined by the (recurrence) properties of the critical orbit . For this reason, we consider and we will analyze the returns of the critical orbit to the *central interval*

and its neighborhood where , . (Observe that these definitions make sense because the fixed points and satisfy for .)

Definition 2A point is -regular if there exists and an open interval such that and is a diffeomorphism.

Definition 3A parameter is regular if most points in are regular, i.e., there are constants and such that

for all . The set of regular parameters is denoted by .

As we already said, our current goal is to show the following result:

Theorem 4If is a regular parameter, then there exists such that

for Lebesgue a.e. , and admits an invariant probability measure which is absolutely continuous with respect to the Lebesgue measure.

** 4.2. The “tent map” parameter **

Before proving Theorem 4, let us warm-up by showing that the parameter is regular and, moreover, satisfies the conclusions of this theorem.

The polynomial is conjugated to the tent map

This conjugation can be seen by interpreting as a Chebyshev polynomial:

The hyperbolic features displayed by (described below) are related to the fact that the critical point is pre-periodic: and for all , so that the critical orbit does not enter the central interval (and, thus, it is not recurrent).

The relationship between and the tent map allows us to organize the returns of the orbits of to as follows. Let and define recursively and as

Note that the interpretation of as a Chebyshev polynomial says that

In particular, is a decreasing sequence converging to and is an increasing sequence converging to .

In terms of these sequences, the return map of to is not difficult to describe:

- recall that the critical point does not return to (because for all );
- the points at return after one iteration: ;
- for , the return map on the intervals and is precisely ; moreover, is an orientation-preserving, resp. orientation-reversing, diffeomorphism from , resp. , onto , and , ;
- for , extends into diffeomorphisms from neighborhoods of to .

The graph of the return map of to looks like two copies of the Gauss map (where denotes the fractional part): it is an instructive exercise to draw the graph of .

At this point, we are ready to establish the regularity of .

Proposition 5The parameter is regular.

*Proof:* From the previous description of the return map , we see that, for , the set of -regular points is precisely

Therefore,

so that is a regular parameter because, for and some constant , one has

for all .

Next, we show that satisfies the conclusion of Jakobson theorem.

Proposition 6preserves the probability measure on and

for Lebesgue a.e. .

*Proof:* The reader can easily check (either by direct computation or by inspection of the conjugation equation between and the tent map) that

for all and with ). It follows that preserves the probability measure on .

So, our task is reduced to compute the Lyapunov exponent of . Since Lebesgue a.e. eventually enters the central interval and the normalized restriction is absolutely continuous with respect to the Lebesgue measure, it suffices to prove that

for -a.e. .

Denote by the first return time of , so that (by definition).

Note that is an ergodic -invariant probability measure such that

Therefore, is a -integrable function, so that Birkhoff’s theorem says that

This asymptotic information on the return times permits to compute the Lyapunov exponent of as follows.

Since the -orbit of a point is confined to and , we see from the chain rule that

for all . By combining this estimate with (2), we obtain

for -a.e. .

In other words, it suffices to compute the Lyapunov exponent along the subsequence of return times. As it turns out, this task is not difficult. Indeed, from (1), we have

Since for , we conclude that

as . This completes the proof of the proposition.

** 4.3. Dynamics of when is regular **

We shall use the same ideas from the previous subsection to show that satisfies the conclusion of Theorem 4 when is regular. More precisely, we will use the returns to of certain “maximal” intervals of regular points to describe the return map of as an uniformly expanding countable Markov map. As it turns out, one can easily construct a -invariant absolutely continuous probability (via the classical approach of thermodynamical formalism [i.e., analysis of Ruelle transfer operator]), and our regularity assumption on permits to convert into a -invariant absolutely continuous probability measure (by a standard summation procedure).

Definition 7An interval is regular of order if there exists a neighborhood of such that is a diffeomorphism from to with . We denote by the inverse branch of the restriction of to .

Remark 6By definition, all points in a regular interval of order are -regular.

As we are going to see in a moment, the definition of regular interval of order was setup in a such a way that the “margin” provided by ensures good distortion and expansion properties of .

Remark 7In a certain sense, the previous paragraph is the real analog of Koebe distortion theorem and Schwarz lemma ensuring good properties for univalent maps in terms of the modulus of certain annuli.

**4.3.1.** **Distortion estimates**

The Schwarzian derivative measures how far is a diffeomorphism from a Möbius transformation, and, for this reason, it is extremely efficient for studying distortion effects on derivative.

In fact, the composition rule reveals that the iterates have negative Schwarzian derivative on any regular interval of order (because has negative Schwarzian derivative).

In particular, if is the inverse branch of the restriction of to the neighborhood a regular interval of order , then has positive Schwarzian derivative on .

Since a diffeomorphism with positive Schwarzian derivative satisfies the estimate (see, e.g., Exercise 6.4 at page 166 of this book of de Faria-de Melo), we conclude:

Lemma 8 (Bounded distortion estimate)For any regular interval , the inverse branch satisfies

for all . In particular,

and, for all measurable subset ,

where .

**4.3.2. Expansion estimates**

Let be the countable family of regular intervals of positive order which are maximal for these properties.

Given , the composition is the inverse branch associated to a regular interval . Conversely, *all* regular intervals contained in are obtained in this way.

The bounded distortion estimates above allow to deduce uniform expansion estimates for the first return map of on regular intervals.

Lemma 9 (Expansion estimates)Let be a regular interval of positive order contained in and denote . Then,

and

where

*Proof:* The first inequality follows from Lemma 8 by setting and . The second estimate follows from the first inequality after taking into account the definition of . Finally, the third estimate follows from the second inequality, the distortion estimate in Lemma 8 and the fact that (so that , and, hence, for some ).

**4.3.3. Thermodynamical formalism for regular parameters **

Let be a regular parameter. By definition,

for some constants and all . In particular, if

then . Thus, most of the dynamics of the return map is described by ,

where is the order of .

Note that is a countable uniformly expanding Markov map (thanks to Lemmas 8 and 9). Using this fact, we will construct a -invariant absolutely continuous probability measure.

For this sake, we consider the dual of the pullback operator acting on finite absolutely continuous measures on . More concretely, the (Ruelle operator) acts on densities by the formula

derived from the change of variables formula and the definition .

We will take advantage of tools from Complex Analysis in order to study .

More precisely, we want to use the standard procedure (from the usual proof of Bogolyubov-Krylov theorem) of building invariant measures as the limit of Cesaro means

where is the constant function on taking the value one. Thus, our task is to show the convergence of the sequence (and this amounts to establish adequate estimates on ).

In this direction, let us show that admits an holomorphic extension. Let be the simply connected domain . Since the critical values of are all real (for all and ), the inverse branches associated to regular intervals extend to univalent maps satisfying

Given a regular interval, let be the sign of on and define

The family is normal on because are univalent functions on such that and is uniformly bounded (by the distortion estimate in Lemma 8).

Since for all , the series

defines a holomorphic extension to of the functions

Moreover, the distortion estimates in Lemma 8 say that

Therefore, the family , , is normal (because belongs to the closed convex hull of the normal family ) and we can extract a subsequence of converging uniformly on compact subsets of to a holomorphic function on such that

Lemma 10is an ergodic -invariant measure equivalent to Lebesgue measure with total mass .

*Proof:* By (4), is a measure equivalent to the Lebesgue measure. Since preserves the total mass,

and is uniformly bounded (thanks to (3)), we obtain that is a -invariant measure with total mass .

Finally, the ergodicity of is proved by the following standard argument (exploiting the good distortion and expansion properties of ). Suppose that is a -invariant subset of positive , or equivalently Lebesgue, measure. Fix a Lebesgue density point and, for each , denote by the regular interval containing . By Lemma 9, we know that (exponentially fast) as . Hence, given , there exists such that

for all . By plugging this information into the distortion estimate in Lemma 8 for , , and by using the -invariance of , we get

Since is arbitrary, we conclude that as desired.

**4.3.4. Lyapunov exponent of for
**

For any , the return time function and the logarithm of the derivative of the return function are -integrable with respect to for all . Indeed, the regularity of the parameter means that

and hence the -integrability of follows from the fact that is equivalent to the Lebesgue measure (cf. Lemma 10). Also, the expansion estimate in Lemma 9 says that is bounded from below (by ), while the fact that for all says that , so that the -integrability of follows from the corresponding fact for .

By Birkhoff theorem, we have (i.e., Lebesgue) almost everywhere convergence of the Birkhoff sums

where .

Note that (because for all ), (by Lemma 9) and

Let us compute the Lyapunov exponent of using (6) and the same argument from Section 4.

Proposition 11Let be a regular parameter. Then, for Lebesgue almost every , one has

*Proof:* Recall that, for every and , we have

Since (5) implies that as , the desired proposition is a consequence of the estimate above and (6).

**4.3.5. Absolutely continuous invariant measures for ,
**

Let us use the invariant measure of the acceleration of to build an invariant measure of on .

The basic idea is very simple: we want to use the iterates of between the initial time and the return time to produce from . More concretely, given a continuous function on , let

Remark 8is well-defined because and for all .

Note that has total mass (by Lemma 10). Moreover, is supported on

that is

for . Furthermore, is -invariant is a consequence of the -invariance of because differs from by a coboundary:

Also, is -ergodic: any -invariant function restricts to a -invariant function on ; hence, the ergodicity of implies that is almost everywhere constant, and so is almost everywhere constant on .

Finally, the formula (7) defining shows that it is absolutely continuous with respect to the Lebesgue measure and its density is given by

This completes the verification of the conclusion of Jakobson theorem for regular parameters .

Closing this post, let us observe that the density of is not square-integrable.

Remark 9By inspecting the terms , and in the definition of , we see that the distortion estimates (4) imply that for almost every ,

for almost every and

for almost every . Therefore, is bounded away from zero but is not in .

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**1. Introduction **

A classical problem in Dynamical Systems is the investigation of closed trajectories in billiards in polygons.

In the case of rational polygons (i.e., polygons whose angles belong to ), it is known that closed trajectories are abundant. A popular way to establish this fact passes through the procedure of unfolding a rational polygon into a flat surface: roughly speaking, instead of letting the trajectories reflect on the boundary of the polygon, we reflect (finitely many times) the boundaries of the polygon in order to obtain straight line trajectories on a flat surface. See this excellent survey of Masur-Tabachnikov for more details.

For our purposes, we recall that a *flat surface* is the data of a Riemann surface , a non-trivial holomorphic -form and the flat metric thought as the Kähler form . Note that has zeros, i.e., has (conical) singularities, whenever has genus (by Riemann-Roch theorem).

Some of the key features concerning closed trajectories in flat surfaces are:

- closed geodesics of the flat metric come in families of parallel trajectories called
*cylinders*in the literature; - such closed geodesics occur in a dense set of directions in the unit circle ;
- Eskin and Masur showed the following asymptotics for the problem of counting cylinders: there exists a constant such that the number of (cylinders of) closed trajectories of length is .

The goal of this post is to generalize this picture to higher dimensions or, more precisely, to K3 surfaces.

**2. K3 surfaces **

Definition 1A compact complex -dimensional manifold is a K3 surface if

- (i) admits a (global) nowhere vanishing holomorphic -form ;
- (ii) the first Betti number is zero (this is equivalent to in this context).

Two basic examples of K3 surfaces are:

Example 1Quartic surfaces in , i.e., , is a polynomial of degree .

Example 2 (Kummer examples)Let be a complex torus (i.e., is a lattice of ). Then, has a subset of singular points, and the blow-up is a K3 surface.

Some basic properties of K3 surfaces include:

- all K3’s are diffeomorphic;
- all K3’s are Kähler (Siu);
- has rank , the Hodge intersection form has signature on , and is an even unimodular lattice;
- the Hodge decomposition of is , where has rank and signature , has rank and signature , and has rank and signature ;
- the data in the previous two items determine the K3 surface (by Torelli theorem).

See, e.g., the lecture notes of D. Huybrechts for more details.

**3. Special Lagrangian submanifolds **

The natural generalization of closed trajectories on flat surfaces are *special Lagrangian submanifolds* (SLags).

Definition 2Let , is a compact complex -dimensional manifold, is a Kähler form, is a holomorphic -form. A real -dimensional submanifold is a special Lagrangian submanifold (SLag) if

- is Lagrangian, i.e., ;
- is special, i.e., .

Remark 1Special Lagrangians are minimal submanifolds (in the sense that they minimize the volume in their cohomology class).

The next example justifies the claim that special Lagrangian submanifolds are the analog of closed trajectories on flat surfaces.

Example 3Consider the case , i.e., is a flat surface. In this situation, all real -dimensional submanifolds are Lagrangian. On the other hand, since is locally (away from its divisors) in this setting, we see that a special Lagrangian is a horizontal geodesic of the flat metric. In particular, if we replace by , then the SLags become the straight line trajectories at angle on .

In a similar vein, special Lagrangian fibrations are the analog of cylinders of closed horizontal trajectories on flat surfaces.

**4. Special Lagrangian fibrations **

Definition 3A fibration of over a real -dimensional base is SLag if its fibers are compact -dimensional SLags submanifolds of . The volume of such a fibration is where is any fiber of .

Remark 2The fibers of SLag fibrations are torii. One can compare this with the Arnold-Liouville theorem saying that the fibers of a fibration of a symplectic manifold by compact Lagrangian submanifolds are necessarily torii. In particular, the base has an integral affine structure whose structural group is the semi-direct product of by .

Remark 3Similarly to the case , a typical K3 surface doesn’t admit a SLag fibration.

K3 surfaces possess a significant amount of relevant structures. For example, a particular case of Yau’s solution to Calabi’s conjecture says that:

Theorem 4 (Yau)Let be a K3 surface equipped with a Kähler form . Then, there exists an unique in the same cohomology class of in such that induces a Ricci-flat metric.

Moreover, K3 surfaces with Ricci-flat metrics are hyperKähler manifolds, i.e., they admit three complex structures such that has the following properties:

- is a Ricci-flat Riemannian metric;
- are complex structures satisfying the usual quaternionic relations: ;
- are compatible with : the forms are closed (i.e. ) for .

Equivalently, we can write the data of a hyperKähler manifold as where and . In this way, we obtain a presentation of K3 surfaces bearing some similarities with our definition of flat surfaces.

**5. Statement of the main result **

In this setting, we change direction of SLag fibrations (in analogy with the case of closed trajectories in flat surfaces) through the notion of *twistor families*. More concretely, we consider the sphere and we denote by

where the fiber is equipped with the complex structure for .

At this point, we are almost ready to state the main result of this post: for technical reasons, we will give an impressionistic statement before explaining the true theorem in Remark 4 below.

Theorem 5 (Filip)Fix a (generic) twistor family . Then, there exists and such that

as .

Remark 4This statement is not quite true in the sense that one should count “equators in the twistor sphere” rather than counting points . Indeed, this is so because if a complex structure admits a SLag fibration at some point in the equator , then one also has SLag fibrations with the same “angle” as varies along the equator .

At this point, Filip started running out of time, and, for this reason, he offered the following sketch of proof of his theorem.

The first step is to reduce to counting elliptic fibrations (i.e., holomorphic fibrations whose fibers are elliptic curves).

The second step is to show that if the twistor family is not too special, then the counting problem reduces to the “linear algebra level” of the rank vector space equipped with a form of signature .

Finally, this last counting problem can be solved through quantitative equidistribution results on for the action of a certain -parameter subgroup on the quotient of the stabilizer in of a null vector in by a lattice (that is, the quotient of the semidirect product of and by).

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