This article is part of the PhD thesis project of Rodolfo (under the supervision of Anton Zorich and myself), which started last September 2016. (In fact, one of my motivations to obtain a “Habilitation à Diriger des Recherches” degree last June 2, 2017 was precisely to be able to formally co-supervise Rodolfo’s PhD thesis project.)

In this (short) blog post, we discuss some aspects of Rodolfo’s solution to Zorich conjecture (and we refer to the preprint for the details).

**1. Statement of Zorich conjecture**

The study of Lyapunov exponents of the Kontsevich-Zorich cocycle (and, more generally, variations of Hodge structures) found many applications since the pioneer works of Zorich and Forni in the late nineties:

- Zorich and Forni described the deviations of ergodic averages of typical interval exchange maps and translation flows in terms of Lyapunov exponents;
- Avila and Forni used in 2007 the positivity of second Lyapunov of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures (among many other ingredients) to show that typical interval exchange transformations and translation flows are weak mixing;
- Delecroix, Hubert and Lelièvre confirmed in 2014 a conjecture of Hardy and Weber on the abnormal rate of diffusion of typical trajectories on -periodic Ehrenfest wind-tree models of Lorenz gases;
- Kappes and Möller completed in 2016 the classification of commensurability classes of non-arithmetic lattices of , , constructed by Deligne and Mostow in the eighties thanks to new invariants coming from Lyapunov exponents;
- etc.

The success of Zorich in describing such deviations of ergodic averages together with many numerical experiments led Kontsevich and him to conjecture that the Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle with respect to Masur-Veech measures are simple (i.e., their multiplicities are ).

Moreover, Zorich had in mind a specific way to establish the Kontsevich-Zorich conjecture: first, he conjectured that the so-called *Rauzy-Veech groups* (associated to a certain combinatorial description of the matrices of the KZ cocycle appearing along typical trajectories for the Masur-Veech measures) are Zariski-dense in the symplectic groups , ; then, he noticed that the works of Guivarc’h-Raugi and Goldsheid-Margulis on the simplicity of Lyapunov exponents for random products of matrices forming a Zariski-dense subgroup could be useful to deduce the “Kontsevich-Zorich simplicity conjecture” from his “Zariski density conjecture”.

After an important partial result of Forni in 2002, Avila and Viana famously established the Kontsevich-Zorich conjecture in 2007. Nevertheless, the arguments of Avila and Viana were slightly different from the scheme outline by Zorich: indeed, as they pointed out in Remark 6.12 of their paper, Avila and Viana avoided discussing the Zariski closure of Rauzy-Veech groups by showing that Rauzy-Veech groups are *pinching* and *twisting*, and that these two properties *suffice* to get the simplicity of the Lyapunov spectrum (i.e., Kontsevich-Zorich conjecture).

Remark 1It is worth to notice that Zariski density implies pinching and twisting, but the converse is not true in general.

In summary, the solution of the Kontsevich-Zorich conjecture by Avila and Viana via the pinching and twisting properties for Rauzy-Veech groups left open Zorich’s conjecture on the Zariski density of Rauzy-Veech groups.

Remark 2Besides giving stronger information about Rauzy-Veech groups (and, in particular, a new proof of Avila and Viana theorem), Zorich’s conjecture has other applications: for example, Magee recently showed that the validity of Zorich’s conjecture implies that the spectral gap / rate of mixing of the geodesic flow on congruence covers of connected components of the strata of moduli spaces of unit area translation surfaces is uniform.

**2. Hyperelliptic Rauzy-Veech groups**

As we already discussed in this blog, Avila, Yoccoz and myself were able to prove Zorich’s conjecture in the particular case of *hyperelliptic* Rauzy-Veech groups by showing the *stronger* statement that such groups contain an *explicit* finite-index subgroup of : roughly speaking, the Rauzy-Veech group is the subgroup of consisting of matrices whose reduction modulo two permute the basis vectors and .

As it turns out, the hyperelliptic Rauzy-Veech groups are associated to one of the three connected components of the so-called *minimal strata* (consisting of translation surfaces of genus with a unique conical singularity of total angle ): in fact, it was proved by Kontsevich and Zorich in 2003 that the minimal strata (of genus ) have three connected components called *hyperelliptic* , *even* spin and *odd* spin .

**3. Rauzy-Veech groups of minimal strata**

As a warm-up problem, we asked Rodolfo to perform numerical experiments with the Rauzy-Veech groups of the odd connected component of the minimal stratum in genus 3 and the even and odd connected components of the minimal stratum in genus 4. In particular, we told him to “compute” the indices of the reductions modulo two of such a Rauzy-Veech group in and .

After playing a bit with the matrices in his computer, Rodolfo announced (among many other things) that the index in of the Rauzy-Veech group of was 28.

This number ringed a bell because (as it is briefly explained here for instance) contains two orthogonal subgroups , resp. , of indices , resp. , consisting of matrices stabilizing a quadratic form with even, resp. odd Arf invariant of representing the reduction modulo two of the symplectic form. In particular, the fact that the number matches the index of suggest the conjecture that Rauzy-Veech groups of , resp. , is the pre-image of , resp. in under the reduction modulo two map .

Once we convinced ourselves about the plausibility of this conjecture, Rodolfo started working on the geometry of the corresponding *Rauzy diagrams* (graphs underlying the structure of the Rauzy-Veech groups) in order to figure out a systematic way of producing many particular matrices generating the desired candidate groups above.

As it turns out, Rodolfo did this in two steps (which occupy most [15 pages] of his preprint):

- first, he exploits the fact that the level two congruence subgroup of (i.e., the kernel of the natural map ) is generated by the squares of certain symplectic transvections to show that the Rauzy-Veech groups of the odd and even components of contain ; for this sake, he exhibits a rich set of loops in Rauzy diagrams inducing appropriate Dehn twists (giving “most” of the desired symplectic transvections).
- secondly, he proves that the reduction modulo two of the Rauzy-Veech groups of the odd, resp. even components of coincides with , resp. , by using the fact that and are generated by orthogonal transvections.

In summary, Rodolfo showed that the Rauzy-Veech groups of the odd and even components of are explicit subgroups of of indices and .

By putting this result together with the result by Avila, Yoccoz and myself for hyperelliptic Rauzy-Veech groups, we conclude that the Rauzy-Veech group of any connected component of a minimal stratum is a finite-index subgroup of and, *a fortiori*, a Zariski-dense subgroup of .

**4. Rauzy-Veech groups of general strata**

Philosophically speaking, a general translation surface of genus “differs” from a translation surface in the minimal stratum because of the relative homology produced by the presence of many conical singularities. In particular, if we “merge” conical singularities of , we should find a translation surface .

From the geometrical point of view, this philosophy was made rigorous by Kontsevich and Zorich in 2003: indeed, they formalized the notions of “merging” and “breaking” zeroes in order to “reduce” the classification of connected components of general strata to the case of connected components of minimal strata!

From the combinatorial point of view, Avila and Viana obtained the following combinatorial analog of Kontsevich-Zorich geometrical statement: we can merge conical singularities until we end up with a component of a minimal stratum in such a way that a “copy” of the Rauzy-Veech group of a component of a minimal stratum shows up inside any Rauzy-Veech group.

Hence, this result of Avila-Viana (or rather its variant stated as Lemma 6.3 in Rodolfo’s preprint) allows Rodolfo to conclude his proof of (a statement slightly stronger than) Zorich’s conjecture: the Rauzy-Veech group of any connected component of any stratum of the moduli space of unit area translation surfaces of genus is a finite-index subgroup of simply because the same is true for the connected components of the minimal stratum .

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By carefully analyzing Freiman’s argument, Cusick and Flahive constructed in 1989 a sequence converging to as such that for all , and, as it turns out, was the largest *known* element of .

In our recent preprint, Gugu and I described the structure of the complement of the Lagrange spectrum in the Markov spectrum near , and this led us to wonder if our description could be used to find new numbers in which are larger than .

As it turns out, Gugu and I succeeded in finding such numbers and we are currently working on the combinatorial arguments needed to extract the largest number given by our methods. (Of course, we plan to include a section on this matter in a forthcoming revised version of our preprint.)

In order to give a flavour on our construction of new numbers in , we will prove in this post that a certain number

belongs to .

**1. Preliminaries**

Let

be the usual continued fraction expansion.

We abbreviate periodic continued fractions by putting a bar over the period: for instance, . Moreover, we use subscripts to indicate the multiplicity of a digit in a sequence: for example, .

Given a bi-infinite sequence and , let

In this context, recall that the classical Lagrange and Markov spectra and are the sets

where

As we already mentioned, Freiman proved that

and Cusick and Flahive extended Freiman’s argument to show that the sequence

accumulating on has the property that for all . In particular,

was the largest known number in .

**2. A new number in **

In what follows, we will show that

Remark 1is a “good” variant of in the sense that it falls in a certain interval which can be proved to avoid the Lagrange spectrum: see Proposition 5 below.

Remark 2Note that , i.e., if we center our discussion at , then is almost 25 times bigger than .

Similarly to the arguments of Freiman and Cusick-Flahive, the proof of Theorem 1 starts by locating an appropriate interval centered at such that does not intersect the Lagrange spectrum.

In this direction, one needs the following three lemmas:

Lemma 2If contains any of the subsequences

- (a)
- (b)
- (c)
- (d)
- (e)
- (f)

then where indicates the position in asterisk.

*Proof:* See Lemma 2 in Chapter 3 of Cusick-Flahive’s book.

Lemma 3If contains any of the subsequences:

- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- (ix)
- (x)
- (xi)

then where indicates the position in asterisk.

*Proof:* See Lemma 1 in Chapter 3 of Cusick-Flahive’s book and also Lemma 3.2 of our preprint with Gugu.

Lemma 4If contains the subsequence:

- (xii)

then where indicates the position in asterisk.

*Proof:* In this situation,

thanks to the standard fact that if

and

with , then if and only if .

As it is explained in Chapter 3 of Cusick-Flahive’s book and also in the proof of Proposition 3.7 of our preprint with Gugu, Lemmas 2, 3 and 4 allow to show that:

does not intersect the Lagrange spectrum .

Of course, this proposition gives a natural strategy to exhibit new numbers in : it suffices to build elements of as close as possible to the right endpoint of .

Remark 3As the reader can guess from the statement of the previous proposition, the right endpoint of is intimately related to Lemma 4. In other terms, the natural limit of this method for producing the largest known numbers in is given by how far we can push to the right the boundary of .Here, Gugu and I are currently trying to optimize the choice of by exploiting the simple observation that the proof of Lemma 4 is certainly not sharp in our situation: indeed, we used the sequence

to bound , but we could do better by noticing that this sequence provides a pessimistic bound because contains a copy of the word (and, thus, , i.e., can’t coincide with on a large chunk when ).

Anyhow, Proposition 5 ensures that

does not belong to the Lagrange spectrum .

At this point, it remains only to check that belongs to the Markov spectrum.

For this sake, let us verify that

By items (a), (b), (c) and (e) of Lemma 2,

except possibly for with . Since

we have

for all . This proves that belongs to the Markov spectrum.

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In this article, we study the complement of the Lagrange spectrum in the Markov spectrum near a non-isolated point found by Freiman, and, as a by-product, we prove that its Hausdorff dimension is

Remark 1Currently, this paper deals exclusively with lower bounds on . In its next version, Gugu and I will include upper bounds on .

In what follows, we present a *streamlined* version of our proof of based on the construction of an *explicit* Cantor set with .

Remark 2W e refer to our paper for more refined informations about the structure of near .

**1. Perron’s characterization of the classical spectra **

Given a bi-infinite sequence and , let

Here,

is the usual continued fraction expansion, and

is the th convergent.

In 1921, Perron showed that the classical Lagrange and Markov spectra and are the sets

where

**2. Freiman’s number **

In 1973, Freiman showed that

In a similar vein, Theorem 4 in Chapter 3 of Cusick-Flahive book asserts that

for all . In particular, is not isolated in .

Remark 3As it turns out, is the largest known number in : see page 35 of Cusick-Flahive book.

In what follows, we shall revisit Freiman’s arguments as described in Chapter 3 of Cusick-Flahive book in order to prove the following result:

Theorem 1Consider the alphabet consisting of the words and . Then,

**3. A standard comparison tool **

In the sequel, we use the following standard comparison tool for continued fractions is the following lemma (cf. Lemmas 1 and 2 in Chapter 1 of Cusick-Flahive book):

- if and only if ;
- .

Remark 4For later use, note that Lemma 2 implies that if and for all , then when is odd, and when is even.

**4. Proof of Theorem 1
**

Similarly to the discussions in Cusick-Flahive book, we shall use the next lemma (extracted from Lemma 2 in Chapter 3 of this book):

Lemma 3If contains any of the subsequences

- (a)
- (b)
- (f)

then where indicates the position in asterisk.

*Proof:* If (a) occurs, then .

If (b) occurs, then Remark 4 implies that

If (f) occurs, then Remark 4 implies that

We shall also need the following fact:

Lemma 4If is a bi-infinite sequence such that

then .

*Proof:* See the proof of Theorem 4 in Chapter 3 of Cusick-Flahive book (especially the last paragraph at page 40).

These lemmas allow us to conclude the proof of Theorem 1 along the following lines.

Proposition 5Given a bi-infinite sequence

where for all and serves to indicate the zeroth position, then

*Proof:* On one hand, Remark 4 implies that

and

and items (a), (b) and (f) of Lemma 3 imply that

for all positions except possibly for with .

On the other hand,

so that for all . This proves the proposition.

At this point, the proof of Theorem 1 is complete: in fact, Proposition 5 and Lemma 4 together imply that

is contained in .

**5. Lower bounds on **

The Gauss map , (where is the fractional part of ) acts on continued fractions as a shift operator:

Therefore, we can use the iterates of the Gauss map to build a bi-Lipschitz map between the Cantor set introduced above and the dynamical Cantor set

Since the Hausdorff dimension is preserved by bi-Lipschitz maps, an immediate corollary of Theorem 1 is:

On the other hand, the Hausdorff dimension was estimated in Subsection 2.2 of this previous post here. In particular, it was shown that:

By putting Corollary 6 and Proposition 7, we conclude the desired estimate

in the title of this post.

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The wide range of applicability of dynamical Cantor sets is partly explained by the fact that several natural examples of Cantor sets are defined in terms of Dynamical Systems: for example, Cantor’s ternary set is

where is .

In some applications of dynamical Cantor sets, it is important to dispose of estimates on their Hausdorff dimensions: for instance, the celebrated work of Bourgain and Kontorovich on Zaremba’s conjecture needs particular types of dynamical Cantor sets with Hausdorff dimension close to one.

For this reason, a considerable literature on this topic was developed. Among the diverse settings covered by many authors, one finds the articles of Bumby, Hensley, …, Jenkinson-Pollicott, Falk-Nussbaum, where the so-called *thermodynamical methods* are exploited to produce approximations for the Hausdorff dimension of Cantor sets defined in terms of continued fraction expansions (i.e., Cantor sets of number-theoretical nature).

In general, the thermodynamical methods quoted above provide a sequence of fast-converging approximations for the Hausdorff dimension of dynamical Cantor sets: for instance, the algorithm described by Jenkinson-Pollicott here gives a sequence converging to at *super-exponential speed*, i.e., for some constants and , where is the Cantor set of real numbers whose continued fraction expansions contain only and .

In particular, the thermodynamical methods give good *heuristics* for the first several digits of the Hausdorff dimension of dynamical Cantor sets (e.g., if we list for and the first three digits of coincide for all , then it is likely that one has found the first three digits of ).

The heuristic bounds provided by the thermodynamical methods can be turned into rigorous estimates: indeed, one of the goals of the recent work of Jenkinson-Pollicott consists into rigorously computing the first 100 digits of .

However, the conversion of heuristic bounds into rigorous estimates is not always easy, and, for this reason, sometimes a *slowly* converging method producing two sequences and of *rigorous* bounds (i.e., for all ) might be interesting for practical purposes.

In this post, we explain a method described in pages 68 to 70 of Palis-Takens book giving explicit sequences converging slowly (e.g., for some constant and all ) towards , and, for the sake of comparison, we apply it to exhibit crude bounds on the Hausdorff dimensions of some Cantor sets defined in terms of continued fraction expansions.

**1. Dynamical Cantor sets of the real line **

Definition 1A -dynamical Cantor set is

where:

- is an expanding -map (i.e., for every ) from a finite union of pairwise disjoint closed intervals to the convex hull of ;
- is a Markov partition, that is, is the convex hull of the union of some of the intervals , and
- is topologically mixing, i.e., for some , for all .

Example 1As we already mentioned, Cantor’s ternary set is a dynamical Cantor set:

where is the affine map .

Example 2Let be a finite alphabet of finite words . The Cantor set

of real numbers whose continued fraction expansions are given by concatenations of the words in is a dynamical Cantor set.In fact, it is possible to construct intervals such that

where and is the Gauss map: see, e.g., this paper here for more explanations.

By definition, a dynamical Cantor set can be inductively constructed as follows. We start with the Markov partition given by the connected components of the domain of the expanding map defining . For each , we define as the collection of connected components of , .

For later use, for each , we denote by

** 1.1. Hausdorff dimension and box counting dimension **

Recall that the Hausdorff dimension and the box-counting dimension of a compact set are defined as follows.

The Hausdorff -measure of is

and the Hausdorff dimension of is

The box-counting dimension is

where is the smallest number of intervals of lengths needed to cover .

Exercise 1Show that .

** 1.2. Upper bound on the dimension of dynamical Cantor sets **

Let be a -dynamical Cantor set associated to an expanding map . Consider the quantities defined by

*Proof:* Let and fix . By definition, there exists such that

for all .

In other words, given , we can cover using a collection of intervals with such that every has length .

It follows from the definitions that, for each , the pre-images of the intervals under form a covering of by intervals of length . Therefore,

for all and , and, *a fortiori*,

for all and .

Hence, if we define , then

for all .

By iterating this argument times, we conclude that

for all .

Thus,

Since is arbitrary, we deduce from the previous inequality that

Because , we get that , and, *a fortiori*, .

** 1.3. Lower bound on the dimension of dynamical Cantor sets **

Let be a -dynamical Cantor set associated to an expanding map . Consider the sequence defined in the previous subsection and fix be a constant such that for all (e.g., certainly works).

Take such that for all and set

Remark 1If is a full Markov map, i.e., for all , then we can choose and .

Consider the quantities given by

*Proof:* Suppose by contradiction that and take .

By definition, , so that for every there is a finite cover of with

Note that any interval of length strictly smaller than

intersects at most one .

Thus, if we define , then each element of the cover of intersects at most one element of . Hence, if we define

then, given any , one has that has *fewer* elements than .

Consider such that for all . From the definitions, for each , we see that is a well-defined cover of such that

Since , we get that

We want to exploit this estimate to prove that

for some . In this direction, suppose that

In this case, the discussion above would imply

a contradiction because on one hand (by definition of ) and on the other hand (by our choice of ).

In summary, we assumed that , we considered an arbitrary cover of with

and we found a cover of with *fewer* elements than such that

By iterating this argument, we would end up with a cover of containing no elements, a contradiction. This proves that .

** 1.4. Slow convergence of towards the Hausdorff dimension **

Let be a -dynamical Cantor set associated to an expanding map . In general, the sequences discussed above converge slowly towards :

*Proof:* The so-called *bounded distortion property* (see Theorem 1 in Chapter 4 of Palis-Takens book or this blog post here) ensures the existence of a constant such that

for all and .

Let and, for all , define

In this setting, we have that

Therefore, , that is,

This proves the proposition.

**2. Hausdorff dimension of Gauss-Cantor sets **

In this section, we apply the previous discussion in the context of Gauss-Cantor sets , that is, the dynamical Cantor sets from Example 2 above.

For the sake of simplicity of exposition, we will not describe the calculation of the sequences and approaching for a general , but we shall focus on two particular examples.

** 2.1. Some bounds on the Hausdorff dimension of **

Consider the alphabet consisting of the words and . The corresponding Cantor set is the Cantor set — denoted by in the beginning of this post — of real numbers whose continued fraction expansions contain only and .

The theory of continued fractions (see Moreira’s paper and Cusick-Flahive book) says that is the dynamical Cantor set associated to the restriction of the Gauss map to , where

Note that the functions and defined on the interval are the inverse branches of .

Remark 2Here, denotes the infinite word obtained by periodic repetition of the block .

By applying to the Markov partition , we deduce that consists of the intervals , , with extremities

The quantities and are not hard to compute using the following remarks. First, is monotone on each (because are Möbius transformations induced by integral matrices with determinant ), so that the values and are attained at the extremities of . Secondly, the derivative of the Gauss map is . By combining these facts, we get that

and

Hence, the sequences and defined in the previous section are the solutions of the equations

and

(Here, we used Remark 1 and the fact that is a full Markov map in order to get the equation of .)

Of course, these equations allow to find the first few terms of the sequences and approaching with some computer-aid: for example, this Mathematica routine here shows that

and

In particular,

Remark 3The approximations and of (obtained from computing with the words in ) are very poor in comparison with the approximation

provided by the thermodynamical method of Jenkinson-Pollicott (also based on calculations with the words in ). In fact, Jenkinson-Pollicott showed in this paper here that the first 100 digits of are

Thus, the first 18 digits of accurately describe (while only the first two digits of and the first digit of are accurate).

** 2.2. Some bounds on the Hausdorff dimension of **

Let us consider now the alphabet consisting of the words and . The theory of continued fractions says that the convex hull of is the interval with extremities and . The images and of under the inverse branches

of the first two iterates of the Gauss map provide the first step of the construction of the Cantor set . In general, given , the collection of intervals of the th step of the construction of is given by

Hence, the interval associated to a string has extremities and .

Similarly as in the previous subsection, we conclude that the quantities and are given by

and

Thus, and are the solutions of

and

Once more, we can calculate the first terms of the sequences and with some computer-aid: this Mathematica routine here reveals that

and

In particular,

Remark 4After implementing of the Jenkinson-Pollicott method with this Mathematica routine here, we get the following approximations for :

The super-exponential convergence of Jenkinson-Pollicott method suggests that

Note that the first two digits of this approximation are accurate (because the first two digits of and coincide).

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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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