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

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

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

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

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

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

Recall that a billiard map preserves the Liouville measure .

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

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

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

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

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

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

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

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

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

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

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

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

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

also has full -measure.

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

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

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

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

** 2.1. Linear Pesin theory **

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

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

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

where is the canonical basis of .

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

where and .

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

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

** 2.2. Non-linear Pesin theory **

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

In this way, is a map fixing such that

where and .

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

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

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

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

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

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

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

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

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

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

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

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

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

** 2.3. Generalized pseudo-orbits **

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

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

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

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

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

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

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

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

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

for all .

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

** 2.4. Coarse graining **

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

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

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

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

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

and, for each ,

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

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

is the point shadowed by .

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

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

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

** 2.5. Inverse theorem **

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

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

Theorem 9If are -recurrent and

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

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

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

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

** 2.6. Bowen-Sinai refinement method **

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

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

First, we start with the collection , where

The s/u-*fiber* of is

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

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

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

Then, we define as the partition induced by the collection

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**1. Introduction **

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

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

**2. Quadratic family **

The *quadratic family* is where and .

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

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

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

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

(where ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(where stands for the Lebesgue measure).

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

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

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

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

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

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

** 4.1. Preliminaries **

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

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

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

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

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

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

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

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

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

** 4.2. The “tent map” parameter **

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

The polynomial is conjugated to the tent map

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

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

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

Note that the interpretation of as a Chebyshev polynomial says that

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

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

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

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

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

Proposition 5The parameter is regular.

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

Therefore,

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

for all .

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

Proposition 6preserves the probability measure on and

for Lebesgue a.e. .

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

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

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

for -a.e. .

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

Note that is an ergodic -invariant probability measure such that

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

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

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

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

for -a.e. .

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

Since for , we conclude that

as . This completes the proof of the proposition.

** 4.3. Dynamics of when is regular **

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

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

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

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

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

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

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

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

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

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

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

for all . In particular,

and, for all measurable subset ,

where .

**4.3.2. Expansion estimates**

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

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

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

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

and

where

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

**4.3.3. Thermodynamical formalism for regular parameters **

Let be a regular parameter. By definition,

for some constants and all . In particular, if

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

where is the order of .

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

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

derived from the change of variables formula and the definition .

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

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

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

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

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

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

Since for all , the series

defines a holomorphic extension to of the functions

Moreover, the distortion estimates in Lemma 8 say that

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

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

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

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

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

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

Since is arbitrary, we conclude that as desired.

**4.3.4. Lyapunov exponent of for
**

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

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

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

where .

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

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

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

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

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

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

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

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

Remark 8is well-defined because and for all .

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

that is

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

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

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

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

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

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

for almost every and

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

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**1. Introduction **

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

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

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

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

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

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

**2. K3 surfaces **

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

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

Two basic examples of K3 surfaces are:

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

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

Some basic properties of K3 surfaces include:

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

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

**3. Special Lagrangian submanifolds **

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

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

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

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

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

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

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

**4. Special Lagrangian fibrations **

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

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

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

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

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

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

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

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

**5. Statement of the main result **

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

where the fiber is equipped with the complex structure for .

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

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

as .

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

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

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

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

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

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How frequent are *thin groups* among *Kontsevich-Zorich monodromies*?

Instead of explaning the meaning of Sarnak’s question in general, we shall restrict ourselves to the case of Kontsevich-Zorich (KZ) monodromies associated to *square-tiled surfaces*.

More concretely, let be a *square-tiled surface* (also called *origamis*) of genus , i.e., is a finite branched covering which is unramified off and is the pullback of on . We have a natural representation

from the group of affine homeomorphisms of to the group of symplectic matrices of the subspace of integral homology classes of projecting to zero under . In this setting, the *Kontsevich-Zorich monodromy* (associated to the -orbit of in the moduli space of translation surfaces) is the image of , i.e.,

(See e.g. these posts here for more background material on square-tiled surfaces.)

By following Sarnak’s terminology, we will say that is a *thin* group if is an infinite index subgroup of whose Zariski closure is

In the particular case of square-tiled surfaces, Sarnak’s question above is related to the following two problems:

- (a) find examples of square-tiled surfaces whose KZ monodromies are thin;
- (b) decide whether the “majority” of square-tiled surfaces in a given connected component of a stratum of the moduli spaces of unit area translation surfaces has thin KZ monodromy (here, “majority” could mean “all but finitely many” or “almost full probability as the number of squares/tiles grows”.)

The goal of this post is to record (below the fold) some discussions with Vincent Delecroix and certain participants of MathOverFlow around item (a).

Remark 1While we willnotgive answers to items (a) and/or (b) in this post, we decided to write it down anyway with the hope that it might be of interest to some readers of this blog: in fact, by the end of this post, we will show the followingconditional statement: if the group generated by the matrices

has infinite-index in , then a certain square-tiled surface of genus answers item (a) affirmatively.

Remark 2Some “evidence” supporting a positive answer to item (b) is provided by this recent paper of Fuchs-Rivin where it is shown that two “randomly chosen” elements (in ) “tend” to generate thin groups.

**1. Faithfulness and thinness **

Let be a square-tiled surface of genus . For the sake of simplicity, suppose that has no non-trivial automorphisms. In this case, the group is naturally identified with a finite-index subgroup of called *Veech group*, and, therefore, we get a representation

The following proposition produces a certificate of thinness for .

Proposition 1Assume that is virtually free and is faithful. Then, has infinite index in .

*Proof:* If is virtually free, is faithful and has finite index in , then the group of higher rank would contain a lattice isomorphic to a free group.

However, this is impossible because higher rank linear groups do not contain lattices isomorphic to free groups: on one hand, a higher rank linear group satisfies Kazhdan’s property (T), so that its lattices also have this property (by Kazhdan’s theorem), and, on the other hand, a free group does not have Kazhdan’s property (T).

This proposition says that a positive answer to item (a) would follow from a positive answer to the following problem:

- (c) find a square-tiled surface of genus (without non-trivial automorphisms) such that the Veech group is virtually free, the image of is Zariski dense in and is faithful.

As we are going to see in the next section, it is not hard to construct explicit examples of square-tiled surfaces (of genus without non-trivial automorphisms) such that the Veech group is virtually free. In other words, a positive solution of (c) is somewhat related to a positive answer to the following problem of independent interest (for the experts on translation surfaces):

- (d) find a square-tiled surface of genus such that the monodromy representation is faithful.

Remark 3If a square-tiled surface has a rational direction of homological dimension one (i.e., the waist curves of maximal cylinders of in a direction of rational slope span a one-dimensional subspace in absolute homology), then isnotfaithful: indeed, this happens because Dehn twists in such a direction act by the identity matrix on (see, e.g., Lemma 5.3 of this paper here), that is, they induce elements of .In particular, since any square-tiled surface of genus always have rational directions of one-cylinder decompositions (by the works of Hubert-Lelièvre and McMullen), it follows that the answer for the analog of item (d) in genus is always negative.

This remark suggests to try to answer items (c) and (d) by looking at square-tiled surfaces of genus with no one-cylinder rational directions. After a computer search (using Sage), Vincent Delecroix found a beautiful example of such a square-tiled surface whose description occupies the next section.

Remark 4For any prescribed , one can construct infinite families of square-tiled surfaces without -cylinders rational directions (via the Hubert-Lelièvre-Kani invariant described in this paper here).

**2. An origami without one-cylinder decompositions **

Consider the square-tiled surface associated to the pair of permutations

The commutator is

so that is a genus square-tiled surface.

The -orbit of consists of four elements. Indeed, this fact can be checked as follows. We recall that:

- the generators and of act on pairs of permutations by the rules and ;
- the pairs of permutations and give rise to the same square-tiled surface.

Therefore, the -orbit of is where is given by the pair of permutations with

As it turns out, the -orbit of accounts for its entire -orbit because

where

and

Remark 5For later use, observe that the matrix acts on pairs of permutations by . In particular, the action of on is completely described by the formulas

where and .

In summary, the -orbit of can be depicted as follows.

Since the cylinder decompositions of in any rational direction described by the horizontal cylinder decomposition of some element of , we have that all cylinder decompositions in rational directions of have exactly three cylinders.

Remark 6It follows from this discussion that has a single cusp (i.e., single -orbit in ). Also, the homological dimension of in the sense of Forni is three. Thus, by the results in this paper of Forni, the Lyapunov spectrum of the Kontsevich-Zorich cocycle over with respect to the Haar measure has the form

Moreover, the Eskin-Kontsevich-Zorich formula for the sum of non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle imply that , i.e.,

Some numerical experiments (with Sage) indicate that and

**3. The affine homeomorphisms of **

The group of affine homeomorphisms of is the stabilizer of in the moduli space of translation surfaces.

It is not hard to see that the subgroup of automorphisms of is trivial. It follows that the elements of are determined by their linear parts in , that is, the natural map

is injective. Hence, is isomorphic to its image under this map.

The group is the finite-index subgroup of consisting of all elements of stabilizing : in the literature, is called the Veech group of .

We saw in the previous section that . Thus, is an index four subgroup of . Furthermore, is a congruence subgroup of level , and the Teichmüller curve has genus zero. Thus, is generated by elliptic and parabolic elements: indeed, one can check that is generated by the following two elliptic matrices

of orders .

The group structure of is provided by the following lemma:

*Proof:* Consider the twelve cones defined by the following properties:

- for each ;
- each , , consists of the convex combinations of positive multiples of the vectors and , where , , , , , and .

A simple calculation shows that

- for each ;
- , , , , , and .

It follows that and play ping-pong with the tables

and

in the sense that and are disjoint subsets of such that

- , ;
- , .

By the ping-pong lemma, we conclude that .

Remark 7The construction of the cones above is similar to the cones studied in Section 2 of this paper of Brav-Thomas.

**4. The Kontsevich-Zorich monodromy of **

The natural representation is called Kontsevich-Zorich cocycle over the arithmetic Teichmüller curve . In the sequel, we will compute the image under of the generators and of .

** 4.1. The relative homology groups of , **

Given , , let us denote by , resp., the relative cycles on consisting of the bottommost horizontal and leftmost vertical sides of the square numbered .

Note that each square of gives a relation , that is,

- , , , , , , , , ;
- , , , , , , , , ;
- , , , , , , , , ;
- , , , , , , , , .

** 4.2. The action of on the relative homology groups **

The matrix takes to , and it acts on the corresponding relative homology groups by the matrix such that

Similarly, the matrix takes to , and it acts on the corresponding relative homology groups by the matrix such that

Finally, exchange and , resp. and , and it acts on the corresponding relative homology groups by the matrices and such that

and

** 4.3. The absolute homology groups of , **

The absolute homology group has a basis where

and

Note that this basis is adapted to the decomposition in the sense that this decomposition corresponds to the partition where and , i.e.,

and

Moreover, it is worth to point out that, in the basis , the matrix of the restriction to of the intersection form is

** 4.4. The action of on the absolute homology group **

The matrices of , and with respect to the basis are

This allows us to compute the images and of the generators and of under the KZ cocycle . Indeed,

so that

For later use, we observe that these formulas give that the non-tautological subrepresentation of takes values

(with respect to the basis of ) at the two generators and of . Moreover, if we denote by , , then the characteristic polynomials and of the matrices and are

and

**5. Zariski density of in **

The matrices and are *Galois-pinching*, i.e., all roots of their characteristic polynomials and are real and simple, and the Galois groups of and have order (that is, the largest possible for symplectic matrices). Furthermore, the splitting fields of their characteristic polynomials are disjoint.

Indeed, these facts follow from the analysis of the following discriminants

and

related to the quadratic subfields of the splitting fields of and : see Proposition 6.14, Remark 6.15 and Proposition 6.16 in this paper here for more details.

By the Zariski density criterion of Prasad-Rapinchuk (see also page 3 of Rivin’s paper or this blog post here), we deduce that:

Proposition 3The Kontsevich-Zorich monodromy is Zariski-dense in .

Remark 8By the main result in this paper here, the Zariski-density of in implies that the Lyapunov spectrum in Remark 6 is simple, i.e.,

**6. Non-faithfulness of the representation **

Once we have Proposition 3 in our toolkit, it is natural to investigate the thinness of . Here, it is tempting to try to use the thinness certificate discussed in Proposition 1 from Section 1 in order to give a positive (partial) answer to Sarnak’s question.

By definition, the faithfulness of would amount to show that has the same group structure of the Veech group (described in Lemma 2), i.e., .

After some numerical experiments with *all* non-trivial words of length on and and *some* non-trivial words of length on and , I thought that could be faithful.

As it turns out, after I asked about the faithfulness of on MathOverflow, Stefan Kohl noticed that is *not* faithful because (a computer-assisted calculation shows that) the kernel of contains a certain non-trivial word of length in and .

Proposition 4 (S. Kohl)The representation is not faithful because

Remark 9The matrices and appearing in my MathOverflow question generate a conjugate of : indeed,

and

where

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Remark 1Still concerning beautiful talks delivered in this conference, I strongly recommend taking a look at this video of Peter Scholze’s talk on cohomology of algebraic varieties: indeed, I think that he was extremely sucessful in communicating his results to a broad audience of non-experts in Algebraic Geometry (such as myself).

This post is a transcription of my notes for Yves’ lecture and, as usual, all errors/mistakes below are my entire responsibility.

**1. Harmonic maps and quasi-isometries **

Definition 1Let and be Riemannian manifolds. An harmonic map is a critical point of Dirichlet energy

Equivalently, is harmonic whenever it satisfies the Euler-Lagrange equation

associated to Dirichlet energy.

Example 1Constant maps, geodesics ( interval) and, more generally, isometries with totally geodesic images are harmonic maps.

The literature dedicated to the questions of existence, uniqueness and regularity of harmonic maps is vast: see for instance these references here. In particular, we know that if is a simply connected non-positively curved Riemannian manifold, then any harmonic map is smooth () and attains the minimum of Dirichlet energy among all maps coinciding with outside a compact subset of .

Definition 2Given , a map between two metric spaces and is a –quasi-isometry whenever

for all .

A conjecture of Schoen-Li-Wang predicts the existence and uniqueness of an harmonic map within bounded distance to any given quasi-isometric self-map from a symmetric space of rank one.

**Conjecture (Schoen-Li-Wang).** Let be a non-compact symmetric space of rank one, i.e., is a hyperbolic space , , or (where is Hamilton quaternion algebra), or (where is Cayley octonion algebra).

Given a quasi-isometry , there exists an unique harmonic map within bounded distance to in the sense that

Remark 2Schoen made this conjecture for . Subsequently, Li-Wang proposed the generalized version of Schoen’s conjecture described above and they proved the uniqueness part of this conjecture.

Remark 3Before these conjectures were formulated, Pansu had already established them for , and .

Remark 4More recently, Markovic solved the case of Li-Wang conjecture and the initial conjecture of Schoen (i.e., the case ) in these papers here and here. Also, Lemm-Markovic confirmed the Li-Wang conjecture for the case , , in this paper here.

The purpose of this post is to discuss the following theorem of Benoist-Hulin establishing a *generalized* version of Schoen-Li-Wang conjecture:

Theorem 3 (Benoist-Hulin)Let and be non-compact symmetric spaces of rank one. Given a quasi-isometry , there exists an unique harmonic map such that

This result brings (at least) two novelties in comparison with previous theorems in the literature because:

- it settles Schoen-Li-Wang conjecture for (the remaining cases of) quasi-isometries of complex hyperbolic spaces, and
- it deals with quasi-isometries between and with different dimensions such as .

**2. Sketch of proof of Benoist-Hulin theorem **

For the sake of exposition, we will sketch the proof of Theorem 3 for quasi-isometries , i.e., **from now on** we will take and .

We begin by noticing that it is sufficient to show the *existence* of an harmonic map within bounded distance from : indeed, as we already told in Remark 2, the uniqueness of follows from the work of Li-Wang.

** 2.1. Regularization of quasi-isometries **

The first step in the proof of Theorem 3 is to *regularize* : by using bump functions, Benoist-Hulin show (with a 2 pages straighforward calculation) that is within bounded distance to a quasi-isometry whose covariant derivatives are bounded on for all (cf. Proposition 3.4 in Benoist-Hulin paper).

In other words, by replacing by if necessary, we can assume that is a quasi-isometry with and for some constant .

** 2.2. Reduction to a priori -estimates **

The second step is to reduce the construction of to an *a priori* -estimate (through a standard *compactness argument*).

More precisely, let us fix an origin , and let us consider the closed balls in . Next, we take the unique harmonic map satisfying the Dirichlet boundary condition on : the map minimizes the Dirichlet energy

among all maps with on . (These facts were proved by Schoen [see here] and Schoen-Uhlenbeck here and here.)

In this context, the existence of in Theorem 3 can be reduced to an uniform estimate on the distances between and :

Proposition 4Suppose that there exists such that

for all . Then, the sequence converges uniformly on compact subsets of to an harmonic map with

Before showing this proposition, we need to recall a (particular case of a) key lemma due to Cheng (see also Lemma 3.3 in Benoist-Hulin paper) allowing to “upgrade” estimates on harmonic maps into bounds on them.

Lemma 5 (Cheng)Let and a harmonic map such that the following estimate holds:

for some . Then, we have the following estimate on at :

Let us now use this lemma to prove Proposition 4.

*Proof:* By assumption, there exists such that for all , and, furthermore, for some . It follows that, given any increasing sequence , we have

for all large enough.

By Cheng’s lemma 5, this estimate leads to the following bound

for all and sufficiently large.

By Arzela-Ascoli theorem, we can find a subsequence converging uniformly on every ball , , to a continuous map with

Moreover, the harmonic maps minimize the Dirichlet energy and, on each ball , their energies are uniformly bounded

thanks to (1). By a compactness theorem of Luckhaus, this implies that the limiting map is harmonic and minimizes the Dirichlet energy.

Finally, the convergence , , of *whole* sequence follows from the convergence along subsequences and the uniqueness theorem of Li-Wang (see Remark 2) ensuring that the harmonic map with is unique.

In summary, Proposition 4 reduces the proof of Theorem 3 to

Theorem 6 (Benoist-Hulin)There exists such that

for all .

The proof of this theorem has two components:

*Boundary estimates*: one estimates for near ;*Interior estimates*: one estimates for far from .

Remark 5For the interior estimates, Benoist-Hulin use a proof by contradiction, i.e., they rule out the situation where as . Nevertheless, a careful inspection of their arguments shows that can be quantitavely bounded in terms of : see Remark 6 below.

** 2.3. Boundary estimates **

The behavior of for near is controlled by the following result:

Proposition 7Then,Let be a smooth map with and for some . Fix and denote by the harmonic map from the closed ball to with on .

for all .

*Proof:* Given , take closest to it, i.e., . Since and , one has

In order to estimate , we will use a barrier function. More precisely, we consider the geodesic passing through and , and we select a point in this geodesic with the following properties:

- is very far from , i.e., for all ;
- .

Since is harmonic, the function is subharmonic. Moreover, this function coincides with the smooth function on . Therefore, the maximum principle tells us that

for all (with equality for ), where is the smooth harmonic function coinciding with on .

In particular, our choice of implies

The barrier function can be controlled thanks to an estimate of Anderson-Schoen. More concretely, the function vanishes on and a short computation (at page 15 of Benoist-Hulin paper) reveals that its Laplacian verifies

In this context, the estimate of Anderson-Schoen (see Proposition 2.4 in Benoist-Hulin paper) asserts that

Therefore, this estimate together with (2) gives

This proves the proposition.

** 2.4. Interior estimates **

The boundary estimate in Proposition 7 says that a point with

For the sake of contradiction, let us suppose that as . Consider the polar exponential coordinates centered at . We will use these coordinates to study and on a ball of radius . More concretely, if denotes the angle between , then we will contradict the triangle inequality

by adapting an idea of Markovic to find in the set

(where is the geodesic connecting to ) such that

In fact, this will be a contradiction to (4) because we will prove that

as for all .

Pictorially, the previous paragraph is summarized in Figure 1 of Benoist-Hulin paper.

Formally, we proceed as follows:

- first, we give upper bounds for when and ;
- secondly, we give upper bounds for when and for all ;
- finally, we give lower bounds on for some .

**2.4.1 Upper bounds on **

for all with .

*Proof:* Consider as above. The triangle of vertices has sides of lengths:

- (by definition of );
- (because and is -quasi-isometric);
- (by our assumption on ).

In particular, the sum of the lengths of the sides of which are adjacent to the angle is much *bigger* than the length of the opposite side to this angle:

By “elementary hyperbolic geometry”, i.e., the fact that the angle is controlled by the Gromov product (i.e., the “excess” in the triangle inequality)

through the relation

(cf. Lemma 2.1 in Benoist-Hulin paper), we deduce from (5) that

This proves the lemma.

**2.4.2 Upper bounds on **

for all with for all (where is the geodesic from to ).

*Proof:* Note that

by our choice of the polar exponential coordinates at .

In order to estimate , we use a lemma of Gauss saying that

for all .

From the previous two estimates and our assumption that , we deduce that

Finally, we control with the aid of Cheng’s lemma. More precisely, because

for all (since and by hypothesis). Thus, Cheng’s lemma 5 ensures that

In summary, the previous estimate show that

because .

**2.4.3 Lower bounds on **

We start our quest for a point with far away from zero by estimating the Lebesgue measure of .

Lemma 10Suppose that and denote by the normalized spherical Lebesgue measure on . Then,

*Proof:* We write where

and

(where is the geodesic path between and ).

For the sake of exposition, we will discuss *only* the estimate of : indeed, the estimate of in Lemma 4.4 of Benoist-Hulin is very similar (up to a minor technicality which is treated with the aid of Cheng’s lemma) and we refer the curious reader to pages 18 and 19 of the original paper for more details.

The key tool to bound is the following Green formula:

The proof of this Green formula uses the subharmonicity of (which is a consequence of the subharmonicity of the distance function of and the harmonicity of ). More precisely, denote by the compact group of isometries of ) fixing , and let be its Haar measure.

Note that acts transitively on each sphere , . Thus, the subharmonic function takes a constant value on , so that

for all . Because , this proves (6).

Next, we observe that for all : in fact,

Therefore, by combining this estimate with (6), we see from the definition of that

i.e.,

. This completes our sketch of proof of the lemma (modulo the verification of the bound ).

Using Lemma 10, we can find with the property that is far away from zero:

*Proof:* Let us set . By Lemma 10,

(because .)

Since any subset of the Euclidean sphere with normalized Lebesgue measure has diameter , for sufficiently large, we can find such that

where is the angle between the vectors such that

Before proceeding further, we need to recall some facts about the relationship between angles of a triangle in a hyperbolic space and Gromov products (cf. Lemmas 2.1 and 2.2 in Benoist-Hulin paper). Let be a geodesic triangle in with vertices and denote by the corresponding angle at . Then,

- (a) ;
- (b) ;
- (c) whenever ;
- (d) if is a -quasi-isometry, then
where is a constant depending only on .

We use these facts as follows. Note that

On the other hand, *if* we can show that

*then* the items (c), (d), (b) and the estimate (7) ensure that

which would complete the proof of the lemma (by taking or ) in view of (8) (because and are independent of ).

In order to check (9), it is sufficient to verify that

thanks to item (d) above. For this sake, we observe that the item (a) and (7) imply that

because . It follows that the desired estimate (10) holds whenever is sufficiently large so that

This proves the lemma.

At this point, the proof of Theorem 6 is complete: indeed, it suffices to put together the estimate (4) with Lemmas 8, 9 and 11 to derive a contradiction with a scenario where as .

We close this post with the following comment about the arguments of Benoist-Hulin.

Remark 6As we already mentioned, the arguments of Benoist-Hulin provide a quantitative bound on in terms of with , . Indeed, the proofs of the lemmas above show that we get a contradiction when the quantities and satisfy the following inequalities:

- ;
- , and

where is the constant provided by item (d) in the proof of Lemma 11.Since these numerical constraints are fulfilled by any for an appropriate (“explicit”) choice of , one gets the following quantitative version of Benoist-Hulin theorem 6: for all .

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This post is a transcription of my notes for Harald’s talk, and, evidently, all mistakes/errors are my responsibility.

**1. Some notations **

We denote by the group of all permutations of .

For , we define their distance by

Definition 1 (Gromov; Weiss)Let be a group. Given and finite, we say that is a -sofic representation whenever

- (a) is an “approximate homomorphism”: for all with ;
- (b) has “few” fixed points for : for all .

We say that a group is sofic if it has -sofic representation for all finite, all (and some ).

Basic examples of sofic groups are: finite groups, amenable groups, etc. In general, it is known that several families of groups are sofic, but it is an important open problem to construct (or show the existence of) non-sofic groups.

The goal of this post is to discuss a candidate for non-sofic group and its connections to Number Theory.

** 1.1. Higman groups **

For , let

The groups and are trivial, and the group is the so-called Higman group.

Remark 1Several statements in this post can be generalized for for all , but for the sake of exposition we will stick to .

Theorem 2 (Helfgott-Juschenko)Then, for every , there exists and a bijection such thatAssume that is sofic.

- (a) is an “almost exponential function”: for all where is a subset of cardinality .
- (b) for all .

Remark 2The existence of functions as above is “unlikely” when is small. More precisely, it is possible to show that there are no bijections satisfying item (a) with and item (b) when is the fifth power of a prime (cf. Remark 4 below for a more precise statement).

In other words, if we could take and in the statement of Theorem 2, the non-soficity of the Higman group would follow.

Unfortunately, the techniques of Helfgott and Juschenko do not allow us to take in Theorem 2, but they permit to control the integer . More concretely, as we are going to see in Theorem 4 below, the integer can be chosen from any fixed sequence which is *thick* in the following sense:

Definition 3A sequence of positive integers isthickif for every there exists such that

for all .

Remark 3It does not take much to be a thick sequence: for example, the sequences and are thick.

As we already announced, the main result of Helfgott-Juschenko is the following improvement of Theorem 2:

Theorem 4 (Helfgott-Juschenko)Then, for every , there exists and a bijection such thatAssume that is sofic and let be a thick sequence.

- (a) for all where is a subset of cardinality .
- (b) for all .

Remark 4The non-soficity of Higman group would follow from this theorem if the bijection , , provided by this statement could be taken so that

- (a*) for all where is a subset of cardinality .
- (b) for all .

Indeed, this is so because Glebsky and Holden-Robinson proved that there is no verifying (a*) and (b).

Remark 5A natural question related to Theorem 4 is: what happens with fewer iterations in item (b)? In this situation, it is possible to use the fact that the group is trivial to show that, for each , there exists such that for any there is no bijection such that

- (a) for all for .
- (b) for all for .

This last remark can be generalized as follows.

Theorem 5 (Helfgott-Juschenko)Let be coprime. Consider the function given by

Then, the equation

has at most solutions .

Remark 6This theorem improves on a result of Glebsky-Shparlinski.

** 1.2. Main ideas in the proofs of Theorems 4 and 5**

As we are going to see in a moment, the key ingredient in the proof of Theorem 4 is the following result of Kerr-Li and Elek-Szabo

Theorem 6Sofic representations of an amenable group are “conjugated”: if , are -sofic representations of with large enough and small enough in terms of a parameter , then there exists a bijection such that

for all and for -almost all (i.e., for at least values of ).

while the idea of the proof of Theorem 5 is to use a finitary version of Poincaré recurrence theorem.

Let us now see how these ingredients are employed by Helfgott-Juschenko in their proofs of Theorems 4 and 5.

**2. Proof of Theorem 4**

** 2.1. Amenable groups and Baumslag-Solitar groups **

Let be a finitely generated group, say , finite. We say that is amenable if there exists a countable sequence of finite sets exhausting (i.e., ) consisting of almost invariant subsets in the sense that:

for all and .

The Baumslag-Solitar groups are

Example 1is amenable.

It follows from its amenability that is a sofic group. As it turns out, one can give a direct proof of this last fact along the following lines. Given , let be an integer such that

(e.g., let be a sufficiently large prime number), and is coprime with .

Define by

(so that ). One can check that is a -sofic representation.

** 2.2. End of proof of Theorem 4 **

Suppose is sofic. Then, the semi-direct product of and is also sofic (because it is an amenable extension of a sofic group). Here, the semi-direct product is defined by letting a generator act on by conjugation as follows: , .

Denote by the set of words of length on , and let be a -sofic representation.

We think of the Baumslag-Solitar group as sitting inside and , and we consider the restriction as an -sofic representation.

By the theorem of Kerr-Li and Elek-Szabo (cf. Theorem 6 above), is “conjugated” to the representation constructed above, i.e., there exists such that, for all ,

for at least values of .

Since is an almost homomorphism, and , we have that

for almost all () values of . Thus,

and

for almost all values of . Furthermore, since and are conjugated, the last equation is also true for .

Therefore, by adjusting the values of on a subset of values of of cardinality , we obtain such that and

for values of , and for all values of . This completes the proof of Theorem 4.

**3. Idea of proof of Theorem 5**

This short section contains an oversimplified argument because we will implicitly assume that the exponential maps are well-defined in . Nevertheless, the discussion below can be adapted to produce an actual proof of Theorem 5.

Let be a generator of , and suppose that

for a positive proportion of values of .

Then, there exists (bounded in terms of the proportion above) such that (1) holds for a both and for a positive proportion of values of : this is an incarnation of Poincaré recurrence theorem.

In other terms, for a positive proportion of values of , one has

By setting , one deduces that the equation

has a set of solutions with positive proportion.

Using Poincaré recurrence theorem once more, we can find such that (2) holds for a both and for a positive proportion of values of .

By writing , we have a positive proportion of values of satisfying the equation

However, this is a contradiction because , and are fixed, so that the previous equation is a polynomial equation on with a bounded number of solutions (and, thus, it can’t be satisfied for a positive proportion of values of ).

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As it turns out, Sophie Morel was kind towards the non-experts in this subject (like myself): indeed, a large part of the talk (see the video here) was introductory, while the more advanced material was delegated to the lecture notes (available here).

In the remainder of this post, I’ll try to summarize some of the topics discussed by Sophie Morel’s talk (using the video in the link above as the main source).

**Disclaimer:** Since I’m not an expert on this subject, all mistakes in this post are my responsibility.

Remark 1If you are in the Oxford area this week, then you will have the opportunity to learn about the main results in Morel’s talk directly from the authors: indeed, Boxer and Scholze (resp.) will give talks tomorrow, resp. Wednesday on their works (see this schedule here of the corresponding 2015 Clay research conference.)

**1. Motivations **

The basic references for this section are Zagier’s text on modular forms and the lecture notes from Deligne’s Bourbaki seminar talk in 1968–1969.

** 1.1. Modular forms **

Let be the upper-half plane, and consider the action of on by homographies:

A modular form of weight and level is a holomorphic function such that

- (i) for all and ;
- (ii) is holomorphic as .

A modular form is called cuspidal if satisfies (i), (ii) and

- (iii) vanishes at infinity.

Note that the condition (i) implies that any modular form is -periodic: for all . In particular, any modular form has a Fourier series

Since as , we see that the condition (ii) is equivalent to

- (ii)’ if

Also, the condition (iii) ( is cuspidal) is equivalent to .

** 1.2. Hecke operators **

The modular forms of weight and level form a finite-dimensional vector space. For each , the Hecke operator

acts on this vector space.

Remark 2This formula gets significantly simpler when is a prime number.

Among the basic properties of Hecke operators, it is worth to mention that and commute, and is cuspidal if the modular form is cuspidal.

These properties suggest the study of the following objects:

Definition 1is called a proper normalized cuspidal modular form if

- is proper (i.e., an eigenvector) for all Hecke operators ;
- the Fourier coefficient in (1) is normalized: .

Remark 3One can show that a proper normalized cuspidal modular form satisfies for all is prime. In other terms, the eigenvalues of the Hecke operators can be read through Fourier coefficients of proper normalized cuspidal modular forms.

** 1.3. Deligne’s proof of Ramanujan-Petersson conjecture **

In this context, the Ramanujan-Petersson conjecture states that the Fourier coefficients (1) of a proper normalized cuspidal modular form of weight satisfy the following bound:

for all prime.

As it is well-known, this conjecture was settled by Eichler-Shimura in the case , and in full generality by Deligne.

The strategy of Deligne was to “reduce” the Ramanujan-Petersson conjecture to Weil conjectures. Very roughly speaking, given a proper normalized cuspidal modular form , we can divide Deligne’s strategy into two steps:

- (1) for each prime , there exists a continuous irreductible representation such that
- is non-ramified at all prime;
- if , the characteristic polynomial of is ;
- appears in the (Betti) cohomology of a (proper, smooth) algebraic variety over .

- (2) the Riemann hypothesis part of Weil’s conjectures (also established by Deligne) provides useful information on the eigenvalues of acting on the cohomology of algebraic varieties, and this can be exploited to establish the Ramanujan-Petersson conjecture because is the sum of eigenvalues of (cf. the expression for the characteristic polynomial of ).

Before proceeding further, let us give some explanations about (1).

The absolute Galois group is the group of automorphisms of the algebraic closure of .

Given a prime number , we can choose an algebraic closure of the field of –adic numbers and a morphism such that the diagram associated to the arrows , , and commute.

In this way, we obtain an embedding where is the group of continuous automorphisms of .

Remark 4The embedding depends on the choice of . In particular, this embedding is only well-defined modulo conjugation (but, as it turns out, this is sufficient for our purposes).

The group is part of an exact sequence

where the arrow is defined by reduction modulo , or, more precisely, by considering a commutative diagram whose arrows are:

- ,
- ,
- , , .

Here, , resp. is the ring of integers of , resp. , is the finite field of elements, and is the algebraic closure of .

The group is topologically generated by a single element, namely, Frobenius endomorphism .

The kernel of the arrow is called inertia group (but its structure is not need for the sake of this post).

Definition 2We say that a representation of is non-ramified at if and only if . In particular, is well-defined modulo conjugation whenever is non-ramified at (cf. Remark 4).

Remark 5The fact that is well-defined modulo conjugation is not a serious issue for the first step of Deligne’s strategy: indeed, we imposed only a condition on the characteristic polynomial of (and this polynomial depends only on the conjugacy class of ).

Closing this subsection, let us outline the construction of the representation as in the first step of Deligne’s strategy in the particular case of proper normalized cuspidal modular forms of weight .

We begin by introducing the modular surface . The modular form is *not* a function on the non-compact Riemann surface (because it is not -invariant): in fact, the modularity condition (i) (with ) implies that is a section of the line bundle of holomorphic differentials on .

Note that the one-point compactification of is an orbifold topologically isomorphic to a sphere. The fact that is holomorphic at infinity (by condition (ii)) implies that can be extended to (i.e., ).

Next, we consider the Hodge decomposition

of the first (Betti) cohomology group of .

In these terms, the Hecke operators admit the following *geometrical* interpretation. For each , we can choose a *finite-index* subgroup and an element such that the action of on (or or ) consists into taking the pullback under the natural arrows

and then taking the “trace of the operator” induced by the arrows

where is the multiplication by . (See, e.g., the subsection 2.3 of this PhD thesis here for more details.)

In particular, a normalized proper cuspidal modular form

gives rise to a non-trivial simultaneous eigenspace of the operators on and, *a fortiori*, , with eigenvalues .

This geometrical interpretation of permits to construct along the following lines.

First, we observe that and are algebraic varieties over in a “canonical way” (because , resp., are moduli spaces of elliptic curves, resp. elliptic curves with extra (“level”) structure).

This implies that, for each prime, we have an (algebraically defined) isomorphism

from to the étale cohomology group of the algebraic variety over . Moreover, this isomorphim behaves “equivariantly” with respect to the Hecke operators .

In particular, the non-trivial simultaneous eigenspace of associated to can be transferred to .

Secondly, we note that this isomorphism is interesting because acts on . Furthermore, since is defined over (and behaves “equivariantly” with respect to this isomorphism), the transferred non-trivial simultaneuous eigenspace of in is *invariant* under this action of .

In summary, the normalized proper cuspidal modular form induces a representation of in a transferred non-trivial simultaneous eigenspace of in , and, as it turns out, this is (more or less) the representation that we were looking for.

** 1.4. Hecke operators from the adelic point of view **

Before closing this introductory section, let us make the following observation about the geometrical interpretation of the Hecke operators.

Our geometrical construction of Hecke operators involved pullback and “trace” operators related to the varieties (for appropriate choices of ).

Nevertheless, for the sake of *generalizing* this discussion, it is helpful to replace by

where is the ring of adeles (with denoting the “finite adeles”),

(with ),

and

The fact that is closely related to comes from the equality

where and .

The key points of this complicated definition of are:

- each operator acts only on the -th component of ;
- this definition can be generalized to other groups (see below).

**2. The works of Boxer and Scholze **

** 2.1. Hecke algebras **

Let us quickly indicate how the discussion of the previous section for can be generalized to , .

We start by introducing

for (where all implied groups are defined by their natural counterparts in ).

Similarly to the case , we have that is a real-analytic variety (given by the disjoint union of two symmetric spaces for ).

In this setting, the Hecke operators are obtained by taking a prime number and an adequate element , and considering the composition of the pullback and trace operators induced on by the natural arrows (inclusion and multiplication by ):

This construction can be slightly generalized by noticing that, if , then we have an action on of the *Hecke algebra*

equipped with the product given by the convolution with respect to the Haar measure with volume one.

Remark 6For , we studied exclusively the action of certain elements , but from now on we will consider the whole action of .

After the works of Satake, we know that is a commutative algebra whose characters correspond to the conjugations classes of under a *canonical* bijection called Satake isomorphism.

For the sake of convenience, we will put “together” the Hecke algebras for by defining the Hecke algebra unramified off as:

** 2.2. Characters of Hecke algebras and Galois representations **

The following two conjectures are part of the so-called Langlands program.

\noindent**Conjecture.** Let be a character such that the corresponding eigenspace in is non-trivial (i.e., is a sort of “normalized proper cuspidal modular form”). Then, there exists a (continuous) Galois representation such that

- is not ramified on (i.e., is trivial on the inertia group );
- corresponds to under Satake’s isomorphism.

Remark 7The reader certainly noticed the similarity between the statement of this conjecture and the item (1) of Deligne’s strategy of proof of Ramanujan-Petersson’s conjecture. Of course, this is not a coincidence, but this conjecture is somewhat “surprising” in comparison with Deligne’s setting because isnotalgebraic when .

\noindent**Conjecture’.** The same conjecture as above is true when is replaced by the *cohomology with torsion* .

The conjecture above was proved to be true by Harris-Lan-Taylor-Thorne. More recently, George Boxer and Peter Scholze proved (independently) the following result

Theorem 3 (Boxer; Scholze)The Conjecture’ is also true.

Remark 8It is known that (a version of) Conjecture’ (for ) implies Conjecture (by taking the limit ). In particular, the methods of Boxer and Scholze are able to recover the theorem of Harris-Lan-Taylor-Thorne.

** 2.3. Some words about the results of Boxer and Scholze **

We stated Conjecture and Conjecture’ for , , but we can also study them for other reductive connected groups such as , etc.

An interesting feature of this generalization of Conjecture/Conjecture’ is that sometimes becomes an algebraic (Shimura)variety (e.g., for , etc.), and this gives us a *hope* of mimicking the arguments from the first section of this post.

Remark 9For example, if is an imaginary quadratic extension of and act on , then the strategy used for doesThe fact that is algebraic isnotsufficient in general to reproduce the strategy employed for in the first section of this post.notproduce a “good” representation attached to a character .In fact, one gets a good representation

onlyafter applying Langlands functoriality principle to “transfer” the problem from to , and then using the strategy for in this new setting.

In summary, even if is algebraic, one can’t apply the strategy for in a simple-minded way in order to deduce Conjecture/Conjecture’.

On the other hand, the fact that is algebraic for certain choices of does not seem to help us in the context of the results of Boxer and Scholze because we know that is not algebraic in their setting.

Nevertheless, Clozel noticed that the non-algebraic varieties usually are strata in the compactification of an algebraic variety. In particular, one can try to exploit this to build the desired representations from characters appearing in .

Let us illustrate this idea of Clozel in the first non-trivial case of the generalization of the Conjecture, i.e., , imaginary quadratic extension (where is a non-algebraic variety of real dimension 3).

We take , so that is algebraic (but not compact). Its Borel-Serre compactification is not algebraic but its boundary has some components associated to parabolic subgroups of such as Levi’s parabolic subgroup

which turns out to be isomorphic to .

In this way, we get two arrows allowing to relate to with the advantage that is algebraic. In particular, this allows to transfer the problem of showing Conjecture/Conjecture’ from non-algebraic settings to algebraic settings (but this is not the end of the history: cf. Remark 9 above!)

At this point, Sophie Morel runned out of time and she decided to conclude her talk by mentionning that after transferring the problem from to as above, an important ingredient in Boxer and Scholze proof of Conjecture’ is the following theorem:

Theorem 4 (Boxer; Scholze)If appears in , then there exists a (cuspidal) in characteristic zero appearing in such that (mod ).

Then, she told that a “one-sentence proof” of this result is the following: one uses a comparison theorem to relate to cohomology groups of affinoid spaces.

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given by

where and is the shift map equipped with the Bernoulli measure .

By Oseledets multiplicative ergodic theorem, the Lyapunov exponents of the random product of and (i.e., the linear cocycle ) are well-defined quantities (depending only on and ) describing the exponential growth of the singular values of the random products

for any -typical choice of .

Moreover, the fact that and are symplectic matrices implies that the Lyapunov exponents are symmetric with respect to the origin, i.e., for each . In other words, the Lyapunov exponents of the symplectic linear cocycle have the form:

In fact, this structure of the Lyapunov exponents of a symplectic linear cocycle reflects the fact that if is an eigenvalue of a symplectic matrix , then is also an eigenvalue of .

A natural *qualitative* question about Lyapunov exponents concerns their *simplicity* in the sense that there are no repeated numbers in the list above (i.e., for all ).

The simplicity property for Lyapunov exponents is the subject of several papers in the literature: see, e.g., the works of Furstenberg, Goldsheid-Margulis, Guivarch-Raugi, and Avila-Viana (among *many* others).

Very roughly speaking, the basic philosophy behind these papers is that the simplicity property holds whenever the monoid generated by and is *rich*. Of course, there are several ways to formalize the meaning of the word “rich”, for example:

- Goldsheid-Margulis and Guivarch-Raugi asked to be Zariski-dense in ;
- Avila-Viana required to be
*pinching*: there exists whose eigenvalues are all real with distinct moduli; such a is called a*pinching matrix*;*twisting*: there exists a pinching matrix and a*twisting matrix*with respect to in the sense that for*all*isotropic -invariant subspaces and*all*coisotropic -invariant subspaces with .

Of course, these notions of “richness” of a monoid are “close” to each other, but they *differ* in a subtle detail: while the Zariski-density condition on is an *algebraic* requirement, the pinching and twisting condition on makes *no* reference to the algebraic structure of the linear group .

In particular, this leads us to the main point of this post:

How the *Zariski-density* and *pinching and twisting* conditions relate to each other?

The first half of this question has a positive answer: a Zariski-dense monoid is also pinching and twisting. Indeed:

- (a) a modification of the arguments in this blog post here (in Spanish) permits to prove that any Zariski-dense monoid contains a pinching matrix , and
- (b) the twisting condition on a matrix with respect to a pinching matrix can be phrased in terms of the non-vanishing of certain (isotropic) minors of the matrix of written in a basis of eigenvectors of ; thus, a Zariski-dense monoid contains a twisting matrix with respect to any given pinching matrix.

On the other hand, the second half of this question has a negative answer: we exhibit below a pinching and twisting monoid which is not Zariski dense.

Remark 1The existence of such examples of monoids is certainly known among experts. Nevertheless, I’m recording it here because it partly “justifies” a forthcoming article joint with Artur Avila and Jean-Christophe Yoccoz in the following sense.

The celebrated paper of Avila-Viana quoted above (on Kontsevich-Zorich conjecture) shows that the so-called “Rauzy monoids” are pinching and twisting (and this is sufficient for their purposes of proving simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle for Masur-Veech measures).

On the other hand, since a pinching and twisting monoid is not necessarily Zariski dense (as we are going to see below), the results of Avila-Viana (per se) can not answer a question of Zorich (see also Remark 6.12 in Avila-Viana paper) about the Zariski density of Rauzy monoids.

In this direction, Artur, Jean-Christophe and I solve (in an article still in preparation) Zorich’s question about Zariski density of Rauzy monoids in the special case of hyperelliptic Rauzy diagrams, and the main example of this post (which will be included in our forthcoming article with Artur and Jean-Christophe) serves to indicate that the results obtained by Artur, Jean-Christophe and myself can not be deduced as “abstract consequences” of the arguments in Avila-Viana paper.

Remark 2The main example of this post also shows that (a version of) Prasad-Rapinchuk’s criterion for Zariski density (cf. Theorem 9.10 of Prasad-Rapinchuk paper or Theorem 1.5 in Rivin’s paper) based on Galois-pinching (in the sense of this paper here) and twisting properties is “sharp”: indeed, an important feature of the main example of this post is the failure of the Galois-pinching property (cf. Remark 4 below for more comments).

**1. A monoid of 4×4 symplectic matrices **

Let be the third symmetric power of the standard representation of . In concrete terms, is constructed as follows. Consider the basis of the space of homogenous polynomials of degree on two variables and . By letting act on and as and , we get a linear map on whose matrix in the basis is

Remark 3The faithful representation is the unique irreducible four-dimensional representation of .

The matrices preserve the symplectic structure on associated to the matrix

Indeed, a direct calculation shows that if , then

where stands for the transpose of .

Therefore, the image is a linear algebraic subgroup of the symplectic group , and the Zariski closure of the monoid generated by the matrices

and

is precisely .

Remark 4Coming back to Remark 2, observe that does not contain Galois-pinching elements of in the sense of this paper here (i.e., pinching elements of with integral entries whose characteristic polynomial has the largest possible Galois group for a reciprocal polynomial [namely, the hyperoctahedral group]) because its rank is . Alternatively, a straightforward computation reveals that the characteristic polynomial of is

and, consequently, the eigenvalues of are

and

In particular, since the characteristic polynomial of always splits, it is never the case that is Galois-pinching.

On the other hand, the element is pinching because its eigenvalues are

Also, the matrix is twisting with respect to . Indeed, the columns of the matrix

consist of eigenvectors of . Thus, is the matrix of in the corresponding basis of eigenvectors of . Moreover, is twisting with respect to if and only if all entries of and all of its minors associated to isotropic planes are non-zero (cf. Lemma 4.8 in this paper here). Finally, this last fact is a consequence of the following exact calculation (see also the numerical approximations) for and its matrix of minors:

and

In summary, the monoid is pinching and twisting, but not Zariski dense in .

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In the sequel, I will transcript my notes from Shen’s talk.

**1. Introduction **

In Real Analysis, the classical Weierstrass function is

with .

Note that the Weierstrass functions have the form

where is a -periodic -function.

Weierstrass (1872) and Hardy (1916) were interested in because they are concrete examples of continuous but nowhere differentiable functions.

Remark 1The graph of tends to be a “fractal object” because is self-similar in the sense that

We will come back to this point later.

Remark 2is a -function for all . In fact, for all , we have

so that

whenever , i.e., .

The study of the graphs of as fractal sets started with the work of Besicovitch-Ursell in 1937.

Remark 3The Hausdorff dimension of the graph of a -function isIndeed, for each , the Hölder continuity condition

leads us to the “natural cover” of by the family of rectangles given byNevertheless, a direct calculation with the family

does notgive us an appropriate bound on . In fact, since for each , we have

Fortunately, we canfor . Because is arbitrary, we deduce that . Of course, this bound is certainly suboptimal for (because we know that anyway).refinethe covering by taking into account that each rectangle tends to be more vertical than horizontal (i.e., its height is usually larger than its width ). More precisely, we can divide each rectangle into squares, say

such that every square has diameter . In this way, we obtain a covering of such that

for . Since is arbitrary, we conclude the desired bound

A long-standing conjecture about the fractal geometry of is:

**Conjecture** (Mandelbrot 1977): The Hausdorff dimension of the graph of is

Remark 4In view of remarks 2 and 3, the whole point of Mandelbrot’s conjecture is to establish the lower bound

Remark 5The analog of Mandelbrot conjecture for the box and packing dimensions is known to be true: see, e.g., these papers here and here).

In a recent paper (see here), Shen proved the following result:

Theorem 1 (Shen)For any integer and for all , the Mandelbrot conjecture is true, i.e.,

Remark 6The techniques employed by Shen also allow him to show that given a -periodic, non-constant, function, and given integer, there exists such that

for all .

Remark 7A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all integer, there exists such that

for all .

The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem 1 by discussing the *particular* case when and is large.

**2. Ledrappier’s dynamical approach **

If is an *integer*, then the self-similar function (cf. Remark 1) is also -periodic, i.e., for all . In particular, if is an integer, then is an invariant repeller for the endomorphism given by

This dynamical characterization of led Ledrappier to the following criterion for the validity of Mandelbrot’s conjecture when is an integer.

Denote by the alphabet . The unstable manifolds of through have slopes of the form

where , , , and

In this context, the push-forwards of the Bernoulli measure on (induced by the discrete measure assigning weight to each letter of the alphabet ) play the role of *conditional measures along vertical fibers* of the unique *Sinai-Ruelle-Bowen (SRB) measure* of the expanding endomorphism ,

where and . In plain terms, this means that

where is the unique -invariant probability measure which is absolutely continuous along unstable manifolds (see Tsujii’s paper).

As it was shown by Ledrappier in 1992, the fractal geometry of the conditional measures have important consequences for the fractal geometry of the graph :

Theorem 2 (Ledrappier)Suppose that for Lebesgue almost every the conditional measures have dimension , i.e.,

Then, the graph has Hausdorff dimension

Remark 8Very roughly speaking, the proof of Ledrappier theorem goes as follows. By Remark 4, it suffices to prove that . By Frostman lemma, we need to construct a Borel measure supported on such that

where . Finally, the main point is that the assumptions in Ledrappier theorem allow to prove that the measure given by the lift to of the Lebesgue measure on via the map satisfies

An interesting consequence of Ledrappier theorem and the equation 1 is the following criterion for Mandelbrot’s conjecture:

Corollary 3If is absolutely continuous with respect to the Lebesgue measure , then

*Proof:* By (1), the absolute continuity of implies that is absolutely continuous with respect to for Lebesgue almost every .

Since for almost every implies that for almost every , the desired corollary now follows from Ledrappier’s theorem.

**3. Tsujii’s theorem **

The relevance of Corollary 3 is explained by the fact that Tsujii found an explicit *transversality condition* implying the absolute continuity of .

More precisely, Tsujii firstly introduced the following definition:

Definition 4

- Given , and , we say that two infinite words are -transverse at if either
or

- Given , , and , we say that two finite words are -transverse at if , are -transverse at for all pairs of infinite words ; otherwise, we say that and are -tangent at ;
- ;
- .

Next, Tsujii proves the following result:

Theorem 5 (Tsujii)If there exists integer such that , then

Remark 9Intuitively, Tsujii’s theorem says the following. The transversality condition implies that the majority of strong unstable manifolds are mutually transverse, so that they almost fill a small neighborhood of some point (see the figure below extracted from this paper of Tsujii). Since the SRB measure is absolutely continuous along strong unstable manifolds, the fact that the ‘s almost fill implies that becomes “comparable” to the restriction of the Lebesgue measure to .

Remark 10In this setting, Barańsky-Barány-Romanowska obtained their main result by showing that, for adequate choices of the parameters and , one has . Indeed, once we know that , since , they can apply Tsujii’s theorem and Ledrappier’s theorem (or rather Corollary 3) to derive the validity of Mandelbrot’s conjecture for certain parameters and .

For the sake of exposition, we will give just a flavor of the proof of Theorem 1 by sketching the derivation of the following result:

Proposition 6Let . If and is sufficiently large, then

In particular, by Corollary 3 and Tsujii’s theorem, if and is sufficiently large, then Mandelbrot’s conjecture is valid, i.e.,

Remark 11The proof of Theorem 1 in full generality (i.e., for integer and ) requires the introduction of a modified version of Tsujii’s transversality condition: roughly speaking, Shen defines a function (inspired from Peter-Paul inequality) and he proves

- (a) a variant of Proposition 6: if integer and , then for some integer ;
- (b) a variant of Tsujii’s theorem: if for some integer , then .

See Sections 2, 3, 4 and 5 of Shen’s paper for more details.

We start the (sketch of) proof of Proposition 6 by recalling that the slopes of unstable manifolds are given by

for , , so that

Remark 12Since , the series defining converges faster than the series defining .

By studying the first term of the expansion of and (while treating the remaining terms as a “small error term”), it is possible to show that if , then

(cf. Lemma 3.2 in Shen’s paper).

Using these estimates, we can find an upper bound for as follows. Take with , and let be such that distinct elements listed in such a way that

for all , where .

From (3), we see that

for all .

Since

for large enough. Indeed, this happens because

- if ;
- if ;
- as , and as (here we used ).

By combining (4) and (5), we deduce that

for all .

Since , the previous estimate implies that

i.e.,

Thus, it follows from our assumptions (, large) that

This completes the (sketch of) proof of Proposition 6 (and our discussion of Shen’s talk).

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