The key *dynamical* idea to give upper bounds on is to show that any sufficiently large element is realized by a sequence whose past or future dynamics lies in the *gaps* of an appropriate horseshoe.

*Qualitately* speaking, this idea is explained by the following lemma.

Lemma 1Fix a horseshoe of a surface diffeomorphism and a height function. For simplicity, let us denote the orbits of by . Denote by

the corresponding Markov and Lagrange spectra.Let a subhorseshoe of and set

where is the Markov value of . Consider such that , and denote by a point with

Then, either or (where and denote the and limit sets of the orbit of ).

*Proof:* By contradiction, suppose that and .

Since and , we can select and such that for all , and . Also, the definitions allow us to take and such that and .

Fix with dense orbit and consider pieces of the orbit of with and .

Consider the pseudo-orbits . By the shadowing lemma, we obtain a sequence of periodic orbits accumulating whose Markov values converge to . In particular, , a contradiction.

In simple terms, this lemma says that an element with is associated to an orbit whose past dynamics (described by ) or future dynamics (described by ) avoids . Thus, there exists such that either the piece of past orbit or the piece of future orbit avoids a neighborhood of in (i.e., one of these pieces of orbit lives in the gaps of ).

Remark 1As it turns out, this qualitative lemma is not sufficient for our purposes and, for this reason, Gugu and I end up using a quantitative version of this lemma (called Lemma 3.1) in our paper.

Once we got some constraints on the dynamics of orbits generating elements of , our strategy to estimate consists in careful choices of and .

For the sake of exposition, let us explain how our strategy yields some bounds for .

Perron proved that any has the form where .

Consider the subhorseshoe (of sequences formed by concatenations of two consecutive 1’s and two consecutive 2’s).

By applying (a quantitative version of) Lemma 1 (cf. Remark 1), one concludes that if , then the past or future dynamics of lives in the *gaps* of .

This means that, up to replacing by , for all sufficiently large:

- either there is an
*unique*extension of giving a sequence whose Markov value in ; - or there are two continuations and of so that the interval is
*disjoint*from the Cantor setassociated to .

(Here, denotes continued fraction expansions.)

Note that this dichotomy imposes *severe* restrictions on the future of because there are not many ways to build sequences associated to . More precisely, we claim that at each sufficiently large step ,

- either we get a forced continuation ;
- or our possible continuations are and .

Indeed, suppose that we have two possible continuations and . If denotes the interval of numbers in whose continued fraction expansion starts by , then the intervals , appear in the following order on the real line:

Thus:

- the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect ; - the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect

so that our continuations are and . Now, we observe that the intervals and , , appear in the following order on the real line:

Hence:

- the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect ; - the continuation is not possible (otherwise would contain and,
*a fortiori*, intersect

so that our claim is proved.

This claim allows us to bound the Hausdorff dimension of

In fact, the claim says that we refine the natural cover of by the intervals by replacing it by a “forced” or by the couple of intervals

Therefore, it follows from the definition of Hausdorff dimension that

Because we can assume that with and , our discussion so far can be summarized by following proposition:

Proposition 2is contained in the arithmetic sum

where for any parameter satisfying (1).

Since the arithmetic sum is the projection , , of the product set , this proposition implies the following result:

Corollary 3where satisfies (1).

The Hausdorff dimension of was computed with high accuracy by Hensley among other authors: one has . In particular,

where verifies (1).

Closing this post, let us show that (1) holds for and, consequently,

For this sake, recall that

where is the denominator of .

Hence, if we set

then the recurrence formula implies that

where .

Because and for all , we have

This completes the argument because .

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In this context, the entries of the continued fraction expansion of an irrational number are related to *cusp excursions* of typical geodesics in the modular surface (i.e., visits to regions for large).

By exploiting this relationship, Vaibhav Gadre analysed cusp excursions on the modular surface to obtain a proof of the following theorem originally due to Diamond–Vaaler:

Theorem 1For Lebesgue almost every , one has

Furthermore, as it is explained in Gadre’s paper here, his analysis of cusp excursions generalize to geodesic flows on complete non-compact finite-area hyperbolic surfaces and to Teichmüller geodesic flows on moduli spaces of flat surfaces.

As the reader can infer from Section 2 of Gadre’s paper, an important ingredient in his investigation of cusp excursions is the *exponential mixing* property for the corresponding geodesic flows.

Partly motivated by this fact, Vaibhav Gadre and I asked ourselves if the exponential mixing result for “Weil–Petersson like” geodesic flows on surfaces obtained by Burns–Masur–M.–Wilkinson could be explored to control cusp excursions of typical geodesics.

In this post, we record lower and upper bounds obtained together with V. Gadre on the depth of cusp excursions of typical “Weil–Petersson like” geodesics.

Remark 1As usual, all errors/mistakes are my sole responsibility.

Remark 2Our exposition follows closely Section 2 of Gadre’s paper.

**1. Ergodic averages of exponentially mixing flows**

Let be a flow on preserving a probability measure .

Suppose that has *exponential decay of correlations*, i.e., there are constants and such that

for all and all “smooth” real-valued observables in a Banach space containing all constant functions (e.g., is a Hölder or Sobolev space).

*Proof:* We write

By -invariance of , we get

The exponential decay of correlations (1) implies that

This proves the lemma.

**2. Effective ergodic theorem for fast mixing flows**

Suppose that is an exponentially mixing flow on (i.e., satisfies (1)).

Fix and denote .

Theorem 3Given , a function such that for each , and a sequence of non-negative functions, we have for -almost every that

for all sufficiently large (depending on ).

*Proof:* Given , let . Since , we get from Lemma 2 that

Therefore,

Consider the sequence and let . From the estimate (2) with and , and and , we get

and

By Borel–Cantelli lemma, the summability of the series for and the previous inequalities imply that for -almost every

and

for all sufficiently large (depending on ).

On the other hand, the non-negativity of the functions says that

for all . Hence,

for all .

It follows from this discussion that for -almost every and all sufficiently large (depending on )

whenever . Because as and for , given , the previous estimate says that for -almost every

for all sufficiently large (depending on and ). This proves the theorem.

**3. Bounds for certain cusp excursions**

Let be a compact surface with finitely many punctures equipped with a negatively curved Riemannian metric which is “asymptotically modelled” by surfaces of revolutions of profiles , , near the punctures: see this paper here for details.

In this setting, it was shown by Burns, Masur, M. and Wilkinson that the geodesic flow on is exponentially mixing with respect to the Liouville (volume) measure , i.e., for each , the estimate (1) holds for the space of -Hölder functions.

In this section, we want to explore the exponential mixing result of Burns–Masur–M.–Wilkinson to study the cusp excursions of .

For the sake of simplicity of exposition, we are going to assume that there is only one cusp where the metric is isometric to the surface of revolution of the profile for and .

Remark 3The general case can be deduced from the arguments below after replacing the geometric facts about the surfaces of revolution of (e.g., Clairaut’s relations, etc.) by their analogs in Burns–Masur–M.–Wilkinson article (e.g., quasi-Clairaut’s relation in Proposition 3.2, etc.).

Given a vector with base point near the cusp, let be the angle between and the direction pointing straight into the cusp of the surface of revolution of . Denote by the collar in around the cusp consisting of points whose -coordinate satisfies .

**3.1. Good initial positions for deep excursions**

Given a parameter , let . The next proposition says that any vector in generates a geodesic making a –*deep* excursion into the cusp in *bounded* time.

Proposition 4If , then the base point of has -coordinate for a certain time where depends only on .

*Proof:* By Clairaut’s relation, the -coordinate along satisfies

for all (during the cusp excursion).

Thus, the value of the -coordinate along is minimized when : at this instant . Since implies that and , the proof of the proposition will be complete once we can bound by a constant . As it turns out, this fact is not hard to establish from classical facts about geodesics on surfaces of revolution: see, for example, Equation (6) in Pollicott–Weiss paper.

**3.2. Smooth approximations of characteristic functions**

Take a smooth non-negative bump function equal to on and supported on such that . Similarly, take a smooth non-negative bump function equal to on and supported on such that .

The non-negative function is a smooth approximation of the characteristic function of :

- is supported on ;
- there exists a constant depending only on such that
- and
- .

**3.3. Deep cusp excursions of typical geodesics**

At this point, we are ready to use the effective ergodic theorem to show that typical geodesics perform deep cusp excursions:

Theorem 5For -almost every and for all sufficiently large (depending on and ), the base point of has -coordinate for a certain time . (Here, denotes any quantity slightly larger than .)

*Proof:* Fix , , . Let be a parameter to be chosen later and consider the function , for (where ).

The effective ergodic theorem (cf. Theorem 3) applied to the functions introduced in the previous subsection says that, for -almost every and all sufficiently large (depending on and ),

On the other hand, by construction, and for a certain constant and for all .

It follows that, for -almost every and all sufficiently large,

If , i.e., , the right-hand side of this inequality is strictly positive for all sufficiently large. Since the function is supported on , we deduce that if then, for -almost every and all sufficiently large, (where ) for some .

By plugging this information into Proposition 4, we conclude that, if

then, for -almost every and all sufficiently large, the -coordinate of is for some time (where is a constant).

This proves the desired theorem: indeed, we can take the parameter arbitrarily close to in the previous paragraph because as and .

**3.4. Very deep cusp excursions are atypical**

Closing this post, let us now show that an elementary argument *à la Borel–Cantelli* implies that a typical geodesic *doesn’t* perform very deep cusp excursions:

Theorem 6For -almost every and for all sufficiently large (depending on and ), the base point of has -coordinate for all times . (Here, denotes any quantity slightly smaller than .)

*Proof:* Let and be parameters to be chosen later, and denote .

By elementary geometrical considerations about surfaces of revolution (similar to the proof of Proposition 4), we see that if the base point of has -coordinate , then the base point of has -coordinate in for all .

Therefore, if we divide into intervals of sizes , then

Since the Liouville measure is -invariant and the surface of revolution of the profile has the property that the volume of the region is , we deduce that

for all . Because we need indices to cover the time interval , we obtain that

We want to study the set . We divide into and . Because , we just need to compute . For this sake, we observe that

and, *a fortiori*, thanks to (3). In particular,

Note that the series is summable when , i.e., In this context, Borel–Cantelli lemma implies that, for -almost every , the -coordinate is for all and all sufficiently large (depending on ). Since as , we conclude that if

then for -almost every , the -coordinate is for all and all sufficiently large (depending on ).

This ends the proof of the theorem: in fact, by letting , we can take arbitrarily close to in the previous paragraph.

Remark 4By Theorems 5 and 6, a typical geodesic enters the region while avoiding the region during the time interval (for all sufficiently large).Of course, the presence of a gap between and motivates the following question: is there anoptimalexponent such that a typical geodesic enters while avoiding during the time interval (for all sufficiently large)?

Remark 5Contrary to the cases discussed in Gadre’s paper, we can’t show that typically there is only one “maximal” cusp excursion in our current setting (of geodesic flows on negatively curved surfaces with cusps modelled by surfaces of revolution of profiles ). In fact, the exponential decay of correlations estimate proved by Burns–Masur–M.–Wilkinson is not strong enough to provide good quasi-independence estimates for consecutive cusp excursions (with the same quality of Lemma 2.11 in Gadre’s paper).

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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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