This article started with a discussion we had in 2009 about some possible extensions of the celebrated work of A. Avila and M. Viana on the Kontsevich-Zorich conjecture, i.e., simplicity of Lyapunov spectrum of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures, to other ergodic -invariant probability measures on moduli spaces of Abelian differentials/translation surfaces.

Of course, we were aware of examples of ergodic -invariant probability measures associated to certain square-tiled surfaces constructed by G. Forni and his coauthor showing that the simplicity of Lyapunov spectrum is simply false in general, that is, for square-tiled surfaces of genus .

On the other hand, some numerical experiments by A. Zorich (with Mathematica), V. Delecroix and myself (with SAGE) seemed to indicate that the examples found by G. Forni were rare in the sense that, after fixing a stratum (i.e., the orders of zeroes) of the Abelian differentials, almost all (say, all but finitely many) square-tiled surfaces generated ergodic -invariant probability measures such that the Lyapunov spectrum of the Kontsevich-Zorich cocycle is simple.

For this reason, we decided to investigate sufficient conditions for the simplicity of Lyapunov spectrum of square-tiled surfaces (or more precisely, the ergodic -invariant probability measures associated to them) with the hope of showing eventually that the simplicity property for almost all square-tiled surfaces in some stratum (of genus ).

Our first attempt (still in 2009) was based on the computation of the Zariski closure of the group of matrices of the Kontsevich-Zorich cocycle (restricted to the symplectic complement of the so-called tautological Lyapunov subspaces). More precisely, this group of matrices is naturally identified to a subgroup of the symplectic group and one could expect, after listing the possible Zariski closed subgroups of at least for low genus (say ), to derive some geometric criterion (say particularly “easy” to apply to square-tiled surfaces) ensuring that a subgroup of has full Zariski closure.

Unfortunately, after some partial success in the case , we failed to produce a reasonable general criterion “easily applicable to square-tiled surfaces” out of the strategy of the previous paragraph (essentially because putting your hands in the Zariski closure of subgroups of is tricky for large ).

After this first (partially successful, partially frustrated) attempt, we started looking in 2010 for simplicity criteria of Lyapunov spectra closer to Avila-Viana work. More precisely, we began to search for a systematic way of verifying the so-called pinching and twisting conditions of Avila-Viana. Very roughly speaking, the pinching condition asks for a matrix with simple real eigenvalues of distinct modulus and the twisting condition asks for a matrix putting the eigenspaces of in general position. Of course, the advantage of Avila-Viana simplicity criterion is that it ensures simplicity without requiring any computation with Zariski closures. Furthermore, the pinching condition is normally easy to check: indeed, if your cocycle has simple spectrum then some matrix is pinching and, “conversely”, if you have a hard time finding some pinching matrix then there is probably a good reason for this to happen and it is likely that your Lyapunov spectrum is not simple. So, this leaves us with the question of checking the twisting condition in a systematic way.

In general, the twisting condition is a subtle linear algebra problem: we want a single matrix putting all eigenspaces of a given pinching matrix at the same time (and this explains why A. Avila and M. Viana got this property in the context of the Kontsevich-Zorich conjecture through a subtle induction scheme).

Fortunately, in the context of square-tiled surfaces, our matrices have integer coefficients. Therefore, it is intuitive that, if the splitting field of a matrix is disjoint from the splitting field of the pinching matrix (in the sense that is the unique common subfield of these splitting fields), then some products of powers of and put all eigenspaces of in general position: indeed, the condition on splitting fields says that does not share invariant subspaces with and this kind of property is not very far from twisting.

As it turns out, it is not so simple to transform this intuition into a theorem, but we managed to do this along the following lines. First, we convert the twisting condition into a combinatorial statement about the completeness of certain graphs. Secondly, we show that the verification of these graphs are complete can be reduced to prove that some related graphs are mixing. Finally, we show the mixing property of these graphs by proving that our graphs have “lots of arrows” (using elementary Galois theory to “spread arrows around”) and by proving that the non-mixing cases of these graphs with lots of arrows imply the existence of common invariant subspaces of and or . Here, this last step follows from a somewhat lengthy combinatorial study of these graphs in Section 4 (occupying 11 pages of our paper).

Anyhow, once we get our Galois-criterion for simplicity of the Lyapunov spectrum of square-tiled surfaces, we apply it for the stratum , that is, the minimal stratum of genus Abelian differentials with a single zero of order .

In fact, our particular choice of for our application is not by chance: in some sense, it is the smallest stratum where the simplicity of Lyapunov spectrum of square-tiled surfaces is not know, and, more importantly, there is a conjectural classification of (-orbits of) square-tiled surfaces by V. Delecroix and S. Lelievre based on numerical experiments by them. Actually, this classification is very close in spirit to the classification of square-tiled surfaces in the minimal stratum in genus by P. Hubert and S. Lelievre, and C. McMullen.

In a nutshell, the conjecture of Delecroix-Lelievre predicts the existence of exactly distinct types of -orbits of square-tiled surfaces in . Furthermore, it is not hard to produce explicit  representatives of each of these orbits. In our paper, we compute (mostly by hand, but with some computer-assisted calculations with polynomials of degree ) the splitting fields of certain matrices of the Kontsevich-Zorich cocycle over each of these types of orbits of square-tiled surfaces. The outcome of this calculation is the following: one can apply the Galois-criterion for these types of orbits if some discriminants of these polynomials of degree are not squares and this last property holds for all but finitely many square-tiled surfaces in these orbits because of Siegel’s theorem (implying that a “reasonable” degree polynomial takes squares as its values only finitely many times).

In summary, from our Galois-criterion of simplicity and Siegel’s theorem, we deduce (unconditionally) that there are infinitely many square-tiled surfaces in with simple Lyapunov spectrum, and, conditionally to Delecroix-Lelievre conjecture, we get that all but finitely many square-tiled surfaces in have simple Lyapunov spectrum.

Closing this post, let us mention that there is little hope of improving our conditional statement above to “all square-tiled surfaces in have simple Lyapunov spectrum“. Indeed, even though we believe that this last statement is true, the fact that we used Siegel’s theorem (to derive our conditional statement) makes that the current best bounds on the size of the eventual “exceptional set” are doubly exponential (i.e., they are numbers of the form where is a bound on the coefficients of the degree polynomials mentioned earlier) simply because the current quantitative versions of Siegel theorem (such as this one by Y. Bilu) provide only doubly exponential bounds (in some sense, this is not a surprise because the bounds are exponential on the “heights” and the “heights” are logarithmic on the coefficients of polynomials). In particular, a simple-minded strategy based on using some computer program to rule out exceptions for square-tiled surfaces with squares for is out of reach.

The workshop was very interesting in several aspects. First, the topics of the talks concerned different research specialities (as you can see from the schedule here), so that it was an excellent opportunity to learn about advances in other related areas. Secondly, the schedule gave sufficient free time so that we could talk to each other. Also, I was happy to meet new people that I knew previously only through their work (e.g., Alex Kontorovich and John Pardon).

In particular, we had two free afternoons on Wednesday and Friday, and I certainly enjoyed both of them: on Wednesday Alex Eskin drove me to the beach and we spent a significant part of the afternoon talking to each other there, and on Friday I went to Getty Center with Sasha Bufetov, Ursula Hamenstadt, Pat Hooper, John Pardon, Federico Rodriguez-Hertz, John Smillie, and Anton Zorich, where, besides classical painters like Monet, Renoir, etc., I saw

• a painting of Fointainebleau forest by Theodore Rousseau (especially appealing to me because I’m living near Fointanebleau since last January…),
• this beautiful painting of Fernand Khnopff, and
• the Dancer sculputure of Paolo Troubetzkoy (little curiosity: Anton Zorich is almost sure that he is a great-grand father of the mathematician Serge Troubetzkoy).

As usual, the talks were very nice (and they will be available at IPAM website here in a near future), and hence I decided to transcript in this post my notes of one of the talks, namely, John Pardon’s talk on his solution of Hilbert-Smith conjecture for 3-manifolds. Of course, the eventual mistakes in what follows are my entire responsibility.

1. Statement of Hilbert-Smith conjecture

In this section, we will quickly review some of the history behind the Hilbert-Smith conjecture. For a more serious reading, we recommend consulting Terence Tao’s notes on Hilbert’s 5th problem (as well as his post here on Hilbert-Smith conjecture).

The 5th problem in the famous list of Hilbert’s problems (stated in 1900) is the following conjecture.

Conjecture (Hilbert’s 5th problem). Let be a locally Euclidean topological group. Then, has a (unique) Lie group structure.

After the works of Gleason (in 1951-1952), Yamabe (in 1953) and Montgomery-Zippin (in 1952), we know that the answer to Hilbert’s 5th problem is yes.

An important step towards the solution of Hilbert’s 5th problem is the following theorem of Gleason and Yamabe:

Theorem 1 (Gleason-Yamabe) Let be locally compact group. Let be an open set containing the identity element . Then, there exists compact and an open subgroup such that is a Lie group.

Remark 1 Another way of stating this theorem is: we have an exact sequence

where is the inverse limit of the family of Lie groups obtained by shrinking towards and “discrete” stands for a discrete group space (cf. Terence Tao’s comment below).

An important corollary of Gleason-Yamabe theorem is:

Corollary 2 is NSS (no small subgroups) if and only if is a Lie group.

In other words, this corollary provides a criterion to recognize Lie groups (and thus it explains the interest of Gleason-Yamabe theorem to Hilbert’s 5th problem). Namely, if has no small subgroups (i.e., there exists a neighborhood of the identity element containing no non-trivial subgroup of ), then is a Lie group.

As it turns out, Hilbert-Smith conjecture is a generalization of Hilbert’s 5th problem where one asks whether Lie groups are the sole (locally compact) groups to act faithfully on manifolds:

Conjecture (Hilbert-Smith). If a (locally compact) group acts faithfully on a manifold (i.e., we have an injective continuous homomorphism ), then is a Lie group.

It is known (see, e.g., this post of Terence Tao for further explanations) that the Hilbert-Smith is equivalent to the following “-adic version”:

Conjecture (Hilbert-Smith for -adic actions). There is no injective continuous homomorphism where is the group of p-adic integers.

It is not hard to convince oneself that the p-adic case is important for Hilbert-Smith conjecture: let generate and, by abusing notation, denote by the corresponding homeomorphism; then, the sequence converges to the identity map . Of course, this occurs because (for ) are small subgroups of (a non-Lie group!).

In summary, the philosophy behind the Hilbert-Smith conjecture is that a compact group acting non-trivially on can not act very close to .

As it is nicely explained in this post of Terence Tao, the reduction of Hilbert-Smith conjecture to the p-adic case uses the following result of M. Newman in 1931:

Theorem 3 (Newman) Let be an open set containing the unit (Euclidean) closed ball . Suppose that has a -action whose orbits have diameter bounded by , i.e., we have an homeomorphism of period (that is, ) such that all -orbits have diameter . Then, the action is trivial.

Proof: A rough sketch of proof goes like this.

Assume that the action is not trivial and consider the map , .

From the fact that -orbits have diameter , one can check that ( is homotopic to the identity map) and thus its degree is .

On the other hand, factors through a map via the natural projection map . Since the projection has degree , it follows that the degree of is , a multiple of .

This contradiction “proves” the theorem.

Using this type of argument, one can also show that:

Theorem 4 Let be a manifold with a metric. Then, there exists such that, if is a compact Lie group acting on with orbits of diameter , then the action is trivial.

After this short discussion of the reduction of Hilbert-Smith conjecture to the p-adic case, let us close this section by pointing out that the general case of Hilbert-Smith conjecture is open. Nevertheless, it was known to be true for low-dimensional manifolds, namely, Montgomery-Zippin showed in 1955 that the conjecture is true for and dimensional manifolds.

In next section, we will discuss the case of -manifolds (after Pardon).

2. Hilbert-Smith conjecture in dimension 3

For the sake of this exposition, let be a connected, orientable, irreducible -manifold with and exactly two ends, e.g., where is a genus surface.

Using the orientation, it makes sense to call one end of the “ end” and the other end of the “ end”.

The basic idea of Pardon to show the Hilbert-Smith conjecture in dimension is to reduce it to a -dimensional problem (that one can handle using our knowledge of the mapping class group of surfaces). In this direction, let be the set of surfaces such that is incompressible (i.e., injects into ) and separates the – end from the + end (i.e, generates ) modulo isotopies (or, equivalently, homotopies).

Definition 5 Given two surfaces , we say that if and only if there are surfaces isotopic to (resp.) such that is contained in the “- end” , that is, is to the “left” of . For example, when , the surface is to the left of .

The first important fact about is:

Lemma 6 is a partially ordered set.

The second (crucial) fact about is the following lemma suggested by Ian Agol to John Pardon:

Lemma 7 (Agol) is a lattice, i.e., for all , the set

has a least element.

Proof: A rough sketch of proof goes as follows. By looking at the figure below

one sees that is a natural choice (where , resp. , is the “+ end”, resp. “- end” of , resp. ).

However, this might not be a good choice because the intersection between and might be “artificially complicated” like in the figure below:

Here, J. Pardon overcomes this difficulty by using the following result of M. Freedman, J. Hass and P. Scott saying that if the representatives and of and minimize area, then the intersection between and is “minimal”:

Theorem 8 (Freedman-Hass-Scott) Let be incompressible. Assume that and are area-minimizing representatives of their homology classes. If and can be isotoped to be disjoint, then and are already disjoint unless they coincide.

Using this result, Pardon shows that is a least element of (where and are area-minimizing representatives of and ).

Remark 2 It is implicit in Pardon’s arguments above that the topological and PL (piecewise linear) categories coincide for -dimensional manifolds (that is, a topological -manifold can be triangulated), a profound theorem of E. Moise. Of course, this result doesn’t extend to higher dimensions and this partly explains why Pardon’s arguments really are “-dimensional”.

At this point, we are ready to give a sketch of proof of Pardon’s theorem:

Theorem 9 (Pardon) There is no injective continuous homomorphism

Proof: Suppose by contradiction that there exists a -action on . Up to replacing by for some large , we can assume that this action is very close to the identity.

Let be a handlebody of genus 2 and denote by its orbit under the -action. Consider now a small arc connecting two boundary points of , denote by its orbit under the -action, and define :

Since acts very close to the identity:

• (1) looks like a handlebody of genus 2 in a coarse scale.

On the other hand, Pardon shows that:

• (2) is non-trivial (here, stands for Cech cohomology).

Now, let be the -neighborhood of (for some -invariant metric) with very small, and define .

By definition, acts on the (invariant) -manifold , and, a fortiori, on the set of incompressible separating surfaces on modulo isotopies.

Since is a lattice (cf. Lemma 7) and a least element of is fixed up to isotopy by the action, we get an action

of on the mapping-class group of .

At this point, we get a contradiction as follows. By item (1), if we look at the projections to of the curves shown in the figure below

we see that contains a submodule fixed by where the intersection is

However, by item (2), the action of on (via ) is non-trivial.

Using these informations, one can deduce the existence of a cyclic subgroup of (essentially the image of under the map ) such that the module annihilated by has a submodule where the intersection form is given by equation (1).

But, Pardon proves that this is a contradiction as follows. Using Nielsen’s classification of cyclic subgroups of the mapping class group (saying that any is realized by a -action on by isometries in some metric), he shows that the intersection form on the module is:

• either
• or

where is the genus of . Thus, there is no submodule of where the intersection form is given by equation (1) and this completes the sketch of proof of Hilbert-Smith conjecture for -manifolds.

• The Second Palis-Balzan International Symposium on Dynamical Systems will be held at Institut Henri Poincaré (Paris, France) from June 10th to June 14th, 2013.
• The Workshop on Combinatorics, Number Theory and Dynamical Systems will be held at IMPA (Rio de Janeiro, Brazil) from August 19th to August 23th, 2013.

The Second Palis-Balzan International Symposium on Dynamical Systems is a part of Project Palis-Balzan – Dynamical Systems, Chaotic Behaviour-Uncertainty, sponsored by the Balzan Foundation, from the prestigious award conferred to Jacob Palis (and IMPA) by the Balzan Foundation in 2010 (with previous winners [in Mathematics category] including A. Kolmogorov, E. Bombieri, J.-P. Serre, A. Borel, M. Gromov and P. Deligne). This project is mainly coordinated by Jacob Palis and Jean-Christophe Yoccoz, and the organizing committee of the Second Palis-Balzan symposium consists of S. Crovisier, J. Palis, Jean-Christophe Yoccoz and myself.

We strongly recommend all potential participants of the Second Palis-Balzan symposium (and especially the ones in the Paris area) to register in the corresponding website: indeed, the staff of Institut Henri Poincaré informed that all conferences will be held in amphithéâtre Hermite whose maximum capacity is 150 persons; thus, it is important for us to have a vague idea of the total number of participants.

The Workshop on Combinatorics, Number Theory and Dynamical Systems is part of a thematic semester on Dynamical Systems to be held at IMPA from August to November 2013. The main organizers of this thematic semester are C. G. Moreira, E. Pujals and M. Viana, and they decided that each month of this semester will be dedicated to a specific topic in Dynamics. In particular, the month of August 2013 will focus on interactions between Combinatorics, Number Theory and Dynamics, and, after some mini-courses, we will have the workshop (organized by C. Mauduit, C. G. Moreira, Y. Lima, J.-C. Yoccoz and myself) mentioned above. Among the confirmed speakers, we have:

• Boris Adamczewski
• Pierre Arnoux
• Tim Austin
• Vitaly Bergelson
• Julien Cassaigne
• Alex Eskin
• Sébastien Ferenczi
• Albert Fisher
• Bryna Kra
• Yoshiharu Kohayakawa
• Ali Messaoudi
• János Pintz
• Miguel Walsh
• Barak Weiss
• Maté Wierdl
• Luca Zamboni

and we expect to confirm the participation of the following mathematicians:

• Jean Bourgain
• Yann Bugeaud
• Hillel Furstenberg
• Elon Lindenstrauss
• Curtis T. McMullen
• Peter Sarnak

I think that this is all I have to say about these conferences for now (but you can look at their respective webpages for updated information). See you! (in Paris or Rio)

• C. Moeglin talked about recent progresses (of J. Arthur) on the discrete spectrum of classical groups,
• G. Malle talked about Ore’s conjecture after Liebeck-O’Brien-Shalev-Tiep,
• J.-B. Gouéré talked about branching Brownian motion, and
• C. Lecuire talked about ending laminations after Brock-Canary-Minsky.

Once more the speakers did a great job in explaining these topics to an audience of non-experts, and, for this reason, I decided to make a post about one of these talk.

Contrary to last time, it was “easy” for me to choose which topic to pick: given my tastes, I had to choose between Ore’s conjecture and ending laminations, and I opted for Ore’s conjecture because the website Images de Mathématiques made available an excellent article (in French) by F. Guéritaud with a guided tour (with plenty of beautiful pictures!) around the works of Masur-Misnky and Brock-Canary-Minsky.

So, below I will transcript my notes of G. Malle’s talk about Ore’s conjecture. As usual, the eventual mistakes in what follows are my entire responsibility.

1. Introduction

Let be a group. Given , we denote by the commutator of and , and we define the commutator subgroup the subgroup of generated by the elements , for . In group theory, is used to define the derived series helping to distinguish certain “categories” of groups such as perfect groups (when ), solvable groups, etc.

By definition, , but, in general, this inclusion is not an equality: for example, there are two (non-isomorphic) groups of order such that the set of commutators is different from (cf. this article of R. Guralnick).

In a more positive tone, N. Ito and O. Ore (independently) showed that:

Theorem 1 (Ito, Ore (1951)) For (symmetric group) and (alternating group), we have the equality .

Furthermore, O. Ore said in his paper that “it is possible that a similar theorem holds for any simple group of finite order, but it seems that at present we do not have the necessary methods to investigate the question”. For this reason, the question of determining whether the previous theorem extends to all finite simple groups became Ore’s conjecture.

In 2010, M. Liebeck, E. O’Brien, A. Shalev and P. Tiep showed in this paper here that Ore’s conjecture is true:

Theorem 2 (Liebeck-O’Brien-Shalev-Tiep (2010)) If is a non-abelian finite simple group, then every is a commutator.

As G. Malle pointed out to us, the proof of Ore’s conjecture by LOST (Liebeck-O’Brien-Shalev-Tiep ) “confirms” O. Ore’s prediction that the tools to attack this problem were not available at the time he wrote his paper. Indeed, LOST’s proof of Ore’s conjecture uses:

• the classification of finite simple groups;
• Lusztig’s parametrization of irreducible characters of finite reductive groups (and, so, Weil conjectures);
• explicit (computer) calculations (approximately 3 years of computer calculations in CPU time)

Concerning the statement of Ore’s conjecture/LOST theorem, let us notice that the result is simply not true for general non-simple groups . For instance, R. Guralnick showed that if and are two finite groups such that is Abelian, , and , then the regular wreath product doesn’t satisfy the equality . (Here, we recall that the wreath product is formed by taking the direct product of several copies of indexed by the elements of and by observing that acts [regularly by left multiplication] on this direct product by permuting the indices, so that it makes sense to talk about the semi-direct product of and ). Also, it is known that the smallest example of a perfect group not fitting the conclusion of Ore’s conjecture/LOST theorem is an extension of (i.e., an elementary Abelian group of order ) by the alternating group .

A question related to Ore’s conjecture and normally attributed to J. G. Thompson is the following one:

Conjecture (J. G. Thompson). Let be a finite non-Abelian simple group. Then, there exists a conjugacy class such that equals to ().

Note that Thompson’s conjecture implies Ore’s conjecture. Indeed, if , then any has the form with . In particular, whenever , its inverse also belongs to : by hypothesis, we can write the identity element as a product for some , so that there is an element whose inverse also belongs to , and, a fortiori, the same is true for all elements of (as they are conjugated to ). Therefore, if we write as with , then, since (as we just saw), we know that there exists such that and, thus, .

Actually, prior to LOST theorem, the proofs of Ore’s conjecture for certain families of groups proceeded by showing that Thompson’s conjecture is true for those families (and this is why we’re mentioning Thompson’s conjecture here). However, this is not the way that LOST’s arguments will work and, in particular, Thompson’s conjecture is still open for the general finite simple group.

2. Direct calculations

The “essential” case in LOST’s proof is that of of Lie type, e.g.,

Indeed, the idea is that for the “remaining” groups (such as the 26 sporadic ones), one can verify Ore’s conjecture by “hands” (or, more precisely, using computers).

Before discussing the case of groups of Lie type, let us consider first the case of simple algebraic groups (that is, the “continuous” analogs of finite groups of Lie type). In this direction, one has the following theorem of S. Pasiencier and H. Wang (over the complex numbers) and R. Ree (over arbitrary algebraically closed fields):

Theorem 3 (Pasiencier-Wang, Ree (1964)) Let be a semisimple linear algebraic group over an algebraically closed field. Then, every is a commutator.

Proof: Given , we have that belongs to some Borel subgroup (by a result of Borel). Denote by Levi’s decomposition of with the (unipotent) radical of and a maximal torus.

In this notation, the proof of the theorem is based on the following two facts:

• (1) For all , there exists a regular element (i.e., an element whose centralizer is ) and an element of the normalizer of such that .
• (2) if is regular, then is a single -conjugacy class.

The first item is due to Kostant and the second item is not hard to deduce (one shows by induction on the central series of that the map from to is bijective when is regular).

In any case, given with Jordan decomposition , and , we use item (1) to select regular and with . Since is regular, we have that is also regular, and, thus, by item (2), there exists such that . It follows that

so that the proof of the theorem is complete.

It is tempting to try to adapt this proof to the case of finite groups of Lie type, but, as it turns out, this is not so easy because it is not true that all elements lie in some Borel subgroup and the analog of item (1) above is simply false.

Nevertheless, E. Ellers and N. Gordeev obtained in these papers here the following result about Gauss decompositions of elements of finite simple groups of Lie type:

Theorem 4 (Ellers-Gordeev (1996)) Let be a finite simple group of Lie type. Denote by a maximally split torus inside a Borel subgroup of with Levi decomposition . Consider the opposite Borel subgroup and let be its Levi decomposition. Fix . Then, for any , there exists such that with and .

The next two corollaries show why this theorem is helpful in our discussion:

Corollary 5 If and denote the -conjugacy classes of two regular elements , then .

Proof: By Ellers-Gordeev theorem, if , then, by taking , we have

On the other hand, by applying item (2) in the proof of Theorem 3 ( is a single -conjugacy class when is regular), we deduce the existence of elements and such that

It follows that

so that the proof of the corollary is complete.

This corollary provides the following criterion for the validity of Thompson’s and Ore’s conjectures for finite simple groups of Lie type:

Corollary 6 If there exists a regular element , then Thompson’s (and, a fortiori, Ore’s) conjecture is true.

Proof: By taking in the previous corollary, we have that any belongs to where is the conjugacy class of .

Using this corollary (among other arguments), E. Ellers and N. Gordeev showed in this paper here the following result:

Theorem 7 (Ellers-Gordeev (1998)) Let be a finite simple group of Lie type over a finite field with . Then, Thompson’s (and Ore’s) conjecture is true.

In view of this theorem, Liebeck-O’Brien-Shalev-Tiep showed their theorem (Ore’s conjecture) by developing a character theory method for the remaining cases that we are going to discuss in the next section.

3. Character theory

Denote by the set of irreducible (complex) characters of . A classical lemma due to Frobenius gives the following criterion to recognize whether an element is a commutator in terms of the characters of :

Lemma 8 (Frobenius) Let be a finite group. Then, is a commutator if and only if

Note that this criterion is suitable for computer calculations: indeed, if we know the character table of , we can ask a computer to determine the sum in order to recognize what elements of are commutators. In particular, this approach works to verify Ore’s conjecture for the 26 sporadic groups (because their character tables are known).

LOST’s idea consists into applying Frobenius lemma in the following way. Firstly, we separate the contribution of the trivial character from other characters

Secondly, we use the classical bound (coming from Schur orthogonality relations) to see that

In particular, the desired sum is not zero if the “error term” in the right-hand side is small, that is, if the conjugacy class is small and/or the dimensions of non-trivial irreducible characters are large.

In order to make this approach work, LOST show the following “Cauchy-Schwarz inequality” controlling the contribution of (non-trivial) irreducible characters with “large” dimension :

Lemma 9 Let be a finite group with conjugacy classes. Then, for all and ,

Of course, a preliminary question before applying this lemma is: what is the size of ? In general, if the finite simple group of Lie type has the form where denotes its rank, then it is known that is bounded by a polynomial in of degree . For example, for , J. Fulman and R. Guralnick proved that

Once we know how to control , the next preliminary question concerns lower bounds on the dimensions of non-trivial irreducible characters (because this gives an idea of the size of the integer for which we will apply the lemma). Here, the Lusztig’s classification of irreducible characters comes into play and one often founds that the following gap phenomenon: there are a few non-trivial irreducible characters close to the smallest possible dimensions and all other non-trivial irreducible characters have dimension at least the square of the smallest possible dimensions. For example, for the symplectic groups , it is possible to show that (cf. this paper of P. Tiep and A. Zalesski):

Lemma 10 Let with odd and . Let be a non-trivial irreducible character of . Then:

• either and is one of the (four) so-called Weil characters of ;
• or .

Proof: The lower bound on is not hard to establish: we can embed into by considering as a -dimensional vector space over ; since has smallest degrees , we get the desired lower bound.

On the other hand, the “gap” result is hard to get: for symplectic groups, one can consult this paper for an elementary proof, but for other types one has to use the full Lusztig’s classification.

After answering the preliminary questions on the sizes of and , we are ready to get back to Ore’s conjecture. By using the information on and , LOST use the estimate on and the “gap” phenomenon above to safely concentrate on the few “Weil characters” with dimensions close to the lower bound. Then, they use explicit values of for a Weil character to show that their lemma works when the size of the -conjugacy class of is small.

In particular, the proof of Ore’s conjecture will be complete if one can show that an element with large conjugacy class is a commutator. In this direction, LOST introduce the concept of breakable elements and they show by induction that breakable elements are commutators. For example, they prove that:

Theorem 11 Let be a breakable element, that is, with . Then, is a commutator.

Here, we said that the proof goes by induction because one can assume that this statement is true for and in order to show the statement for . However, one has to be careful here because some small cases such as require special attention.

Finally, it remains to deal with unbreakable elements with large conjugacy classes. As it is shown by LOST, an unbreakable element that is not covered in their “Cauchy-Schwarz inequality” argument has a really large conjugacy class: for instance, for , they show (sometimes by computing characters) that if is unbreakable, then is small as far as their arguments are concerned. Thus, there is a rather small number of such large conjugacy classes of unbreakable elements, so that one can hope to show that the elements in such a small list of conjugacy classes are commutators by simply exhibiting random commutators belonging to each of them, and this is precisely what LOST do to complete the proof of their theorem.

After presenting this beautiful sketch of proof of Ore’s conjecture/LOST theorem, G. Malle decided to comment on the relationship between Ore’s conjecture and word maps as the last topic of his talk.

4. Word maps

Let denote the free group on generators and let be any group. Given a word , , we define the word map as

We will denote the image of the word map by .

Example 1 For the word , the surjectivity of the word map is equivalent to Ore’s conjecture for .

In 1983, A. Borel showed that the following “almost surjectivity” result for word maps on algebraic groups:

Theorem 12 (Borel (1983)) Let be a semisimple linear algebraic group over an algebraically closed field. Then, for any non-trivial word , the word map is dominant (i.e., its image contains a Zariski open and dense subset of ).

Proof: Firstly, by taking the universal cover of , one can see that it suffices to prove the result when is simple. Secondly, by considering maximal torii, it is possible to further reduce to the case of of type (e.g., if , one reduces the result to the case of ). Finally, one does the type case directly.

Remark 1 Word maps are not surjective in general: for example, in positive characteristic , the image of the word map associated does not contain regular unipotent elements.

This theorem allows to deduce the following result of M. Larsen, A. Shalev and P. Tiep:

Theorem 13 (Larsen-Shalev-Tiep (2011)) Let be two non-trivial words. Then, there exists an integer such that for all finite non-Abelian simple groups with .

As a corollary of this result, we have:

Corollary 14 If , then there exists such that for all (finite non-Abelian simple with) . In particular, .

A short sketch of proof of Theorem 13 goes as follows. We separate the discussion into two cases: groups of bounded rank and groups of unbounded rank.

For the bounded rank case, Larsen-Shalev-Tiep show that:

Theorem 15 (Larsen-Shalev-Tiep) Let and let be an infinite family of finite simple groups of fixed Lie type (e.g., with fixed). Then, there exists such that, for each , the set contains regular elements from any maximal torus of .

Then, they conclude the bounded rank case essentially by combining this theorem with the result of Ellers-Gordeev in Corollary 5 above.

Finally, for groups with unbounded rank, they use that for all finite simple groups of Lie type except type , there are pairs and of maximal torii such that if for some , then is the trivial character or the Steinberg character . At this point, G. Malle ran out of time and thus he decided to finish his talk here.

5. Epilogue

Closing this post, let me mention three question posed to G. Malle after the end of his talk:

• Y. de Cornulier asked about estimates on the number of ways of writing as a commutator. Here, G. Malle mentioned that it is easy to produce lower bounds (essentially by changing the conjugacy classes of the elements you use to write ), but he was not sure about upper bounds.
• J.-P. Serre asked about the nature of computer part of LOST’s proof (i.e., if there were “inner checks” for correctness, etc.) and G. Malle assured him that the computer programs used are not extremely sophisticated (they are based on GAP and Magma) and thus it is unlikely that there are errors coming from the corresponding codes. (Apparently J.-P. Serre was not completely satisfied with this answer as I could deduce from his facial expression…)
• I asked about whether one can write as the commutator of a pair of elements generating (here I had in mind some potential links to origamis/square-tiled surfaces…), and G. Malle told me that, at least for certain families, this is likely to be known (and it is probably contained in the most recent LOST papers), but he was not aware of a reference were the general problem is treated.

Theorem 1 (A. Avila, S. Crovisier and A. Wilkinson) There exists a residual (i.e., -dense) subset such that for any :

• (ZE) either all Lyapunov exponents of -a.e. vanish,
• (NUA) or is non-uniformly Anosov in the sense that
  • has a (global) dominated splitting, i.e., there is a decomposition into -invariant subbundles such that dominates , that is, there exists with for any , unitary vectors (“the largest expansion along is dominated by the weakest contraction in , but, a priori, neither is assumed to be contracted nor is assumed to be expanded”).
  • for -a.e. , the fibers of and of the dominated splitting coincide with the stable and unstable Oseledets subspaces, i.e., and ,

  and is ergodic.

Theorem 2 (A. Avila, S. Crovisier, A. Wilkinson) For , the set of ergodic diffeomorphisms in contains a -open, -dense subset of the set of partially hyperbolic volume-preserving -diffeomorphisms.

Furthermore, we saw a sketch of proof of Theorem 1 based on Sylvain’s talk at LAGA.

Today we’ll focus exclusively on the proof of Theorem 2 based on Sylvain’s talk at Jussieu (assuming, of course, that the reader is familiar with our previous post on this subject).

1. Stable transitivity

Before attacking the problem of stable ergodicity (in -topology) in the statement of Theorem 2, let us discuss the issue of getting the weaker property of stable transitivity.

In this direction, we have the following result:

Theorem 3 There exists a -open, -dense subset of consisting of diffeomorphisms such that the orbits of -a.e. is dense.

As it was explained by Sylvain, this theorem follows from the works of D. Dolgopyat and A. Wilkinson, and K. Burns, D. Dolgopyat, and Y. Pesin.

Indeed, by the results of Dolgopyat-Wilkinson, there exists a -open, -dense subset of consisting of accessible partially hyperbolic diffeomorphisms, that is, partially hyperbolic diffeormorphisms such that any pair of points can be connected by a finite sequence of points whose stable and unstable manifolds are “related” in the sense that for any , and belong to the same stable or unstable leaf.

Intuitively, it is not hard to convince oneself that accessibility has something to do with transitivity/ergodicity: for example, the direct product on of an Anosov diffeomorphism on by the identity map on is a partially hyperbolic diffeomorphism that is not transitive (nor ergodic) in part because the stable and unstable manifolds of are “confined” to the submanifolds , (and this is the “opposite” of the accessibility property).

Using the accessibility property, one can obtain “stable transitivity” (in the sense of Theorem 3) from the following idea of M. Brin (further developed by Burns-Dolgopyat-Pesin here). In order to show that the orbit of -a.e. point of is dense, it suffices to show that, given two open sets and , the orbit of -a.e. passes through . Fix and arbitrarily. By the accessibility property, we can join and with a finite path such that and belong to the same stable of unstable leaf. By using the stable and unstable foliations, we can “propagate” and to construct neighborhoods of saturated by pieces of stable or unstable leaves.

We proceed by induction, that is, we will show that the orbits of -a.e. point of passes through . By Poincaré’s recurrence theorem, -a.e. point is recurrent. Consider the subset of recurrent points. Since is , we can use the absolute continuity of the stable and unstable foliations to see that there exists a subset such that and each point of belongs to the stable or unstable leaf of a point of . Since stable and unstable leaves are uniformly contracted to the future or to the past, we deduce that future or past iterates of accumulate the respective future or past iterates of . Because is recurrent, it follows that the orbit of each visits . Of course, this completes the proof of Theorem 3: starting from , we know that the orbit of -a.e. passes through , …, the orbit of -a.e. passes through , and, thus, the orbit of -a.e. passes through .

After this quick discussion of stable transitivity, let us study a mechanism of stable ergodicity inspired by this article of F. Rodriguez-Hertz, J. Rodriguez-Hertz, A. Tahzibi and R. Ures.

2. -T-U ergodicity criterion

In their article, F. Rodriguez-Hertz, J. Rodriguez-Hertz, A. Tahzibi and R. Ures combine stable transitivity with a “generalized Hopf argument” to prove the following result.

Let be a diffeomorphism with a hyperbolic periodic point . Denote by

and

where and denote the Pesin stable and unstable manifolds of , and the symbol denotes transverse intersection.

Theorem 4 (Rodriguez-Hertz–Rodriguez-Hertz–Tahzibi–Ures) Suppose that and . Then:

• the sets , and coincide modulo subsets of -measure zero, and
• is ergodic.

From this result, the idea to approach Theorem 2 is the following. Given with a splitting into (strong/uniform) stable subbundle , central subbundle and (strong/uniform) unstable subbundle , we have from Theorem 1 that there are -close to ergodic non-uniformly Anosov diffeomorphisms. By some semicontinuity arguments (see Section 2 of the previous post), it is possible to check that, for all -close to :

• ,
• there is a dominated splitting coming from ,
• there exists a hyperbolic periodic point such that , and
• for -a.e. , the eventual zero Lyapunov exponents (if they exist) are all contained in or , i.e., or .

At this point, we observe that, by Theorem 4, if we can show that , then we deduce that the dynamics of all -close to is ergodic.

In other words, the proof of Theorem 2 is reduced to the verification of the equality

Here, a naive strategy to check this equality goes as follows. By Theorem 3, we know that the orbit of -a.e. is dense. In particular, the orbit of passes arbitrarily close to , and we could expect that intersects transversely when (here denotes the complementary invariant subbundle to , or ), i.e., . However, it is known that Pesin stable/unstable manifolds depend measurably on , and, thus, they might get too small to intersect (so that we can’t conclude that )!

In order to overcome this difficulty, we start by observing that, in our context of partially hyperbolic diffeomorphisms, Pesin invariant manifolds can’t be uniformly small in all directions! More precisely, since has a decomposition , we have that contains or , and, a fortiori, contains or .

Evidently, this is still not enough to conclude that because the dimensions of and/or might be “small” compared to (and in this case there is no hope for transverse intersections).

On the other hand, this hints at the following strategy. After the works of Smale, we know that the “measurable homoclinic class” of the hyperbolic periodic point usually contains horseshoes . Moreover, as it was first observed by Christian Bonatti and Lorenzo Diaz, sometimes the horseshoes might work as blenders (in Bonatti-Diaz terminology), that is, the strong unstable manifolds of points collectively behave like a -dimensional submanifold in the sense that any -dimensional submanifold -close to a strong stable leave , , crosses some . In other terms, a blender of unstable dimension is a horseshoe that is “fat” when looked along the strong stable direction . The formal definition of a blender goes as follows:

Definition 5 A horseshoe (i.e., a compact, totally disconnected, locally maximal, hyperbolic set) whose tangent bundle dynamics is is a blender of unstable dimension if there exists a constant and a point such that, if is a -dimensional disk -close to the local strong stable manifold of of size , then intersects for some . (A blender of stable dimension is defined in a similar way)

Of course, this phenomenon reminds Marstrand’s theorem that the projections of fat horseshoes in certain directions might contain positive measure sets, and, in fact, some versions of Marstrand’s theorem and the construction of blenders appear in this recent work of C. G. (Gugu) Moreira and W. Silva.

In any event, by pursuing the ideas described in the previous paragraph, Avila-Crovisier-Wilkinson further reduce the proof of their Theorem 2 to the following statement:

Theorem 6 Given a -diffeomorphism () of a compact manifold such that , there are -diffeomorphisms -close to with a blender of unstable dimension and stable dimension .

In next (and last) section, we will complete the proof of Theorem 2 by sketching the proof of Theorem 6.

3. Construction of blenders

Let be an ergodic component of . By definition, the (ergodic) probability measure is hyperbolic (in Pesin’s sense), i.e., its Lyapunov exponents are all non-zero. Furthermore, since is , Pesin’s formula says that

By some results of A. Katok, given , one can find inside a horseshoe such that:

• its topological entropy is -close to the metric entropy of in the sense that , and
• all invariant measures supported on have Lyapunov exponents -close to .

By the works of F. Ledrappier and L.-S. Young, we can “convert” the informations in the two items above into quantitative information about the fractal dimensions of : very roughly speaking, denoting by the maximal entropy measure of (i.e., ), Ledrappier and Young associated “dimensions” () such that

In our context, we know that is close to , the Lyapunov exponents are close to , so that, by Pesin’s formula, we deduce that the Ledrappier-Young “dimensions” are close to , and, as it is explained in Ledrappier-Young’s works, this means that has large fractal (e.g., Hausdorff) dimension.

Once they dispose of a horseshoe with large fractal dimension, Avila-Crovisier-Wilkinson concentrate their efforts in showing the following theorem:

Theorem 7 Let be a diffeomorphism with a horseshoe with “large fractal dimension”. Then, there are -diffeomorphisms -close to with blenders (of stable and unstable dimensions ).

Of course, Theorem 6 is a consequence of this theorem.

As Sylvain explained to us, the inspiration for this theorem is the following recent result of C. G. Moreira and W. Silva:

Theorem 8 (Moreira-Silva) Let be a -diffeomorphism with a horseshoe with a decomposition with . Then, there are -close to with a blender of unstable dimension .

The attentive reader noticed the following differences between Theorem 7 and Moreira-Silva’s theorem. In the former, there is no assumption on the dimension of , but, in compensation, in the latter one has the stronger conclusion that blenders can be found via -small perturbations.

In fact, Sylvain told that Theorem 7 might also be true with -small perturbations. Indeed, Avila-Crovisier-Wilkinson show Theorem 7 in two steps:

• Step 1: given , there are a subhorseshoe with topological entropy -close to and a -small -perturbation of such that the continuation of is affine, i.e., has a Markov partition , , such that, in adequate coordinates , the return maps of to have the form , where is a matrix and is a constant vector (for each ).
• Step 2: if is an affine horseshoe of with “large fractal dimension”, then there are -perturbations of such that the continuation contains blenders (of stable and unstable dimensions ).

Here, they use -perturbations to place themselves into the favorable situation of affine horseshoes, but, in principle, there is no reason to believe that Step 1 is unavoidable.

Closing his talk, Sylvain said a few words about Steps 1 and 2.

Following him, let us start with Step 2. For the sake of simplicity, let us assume that is a -dimensional manifold, so that our affine horseshoe has a invariant splitting into three 1-dimensional subbundles. In order to get a blender with unstable dimension 2 out of , we can “forget” about the -direction, that is, we will pretend that is an affine horseshoe on a -dimensional submanifold, say is an affine horseshoe living in a square , where the horizontal direction is and the vertical direction is . In this setting, a blender is a horseshoe in intersecting all vertical lines , that is, a horseshoe whose vertical projection contains the interval . Note that a projection doesn’t increase the Hausdorff dimension, it is clear that such blenders have “large fractal dimension” (e.g. for some ). Anyhow, the construction of a horseshoe whose vertical projection contains is not difficult: the basic idea is that this last property is satisfied if the vertical projections of the Markov rectangles defining the horseshoe cover .

In summary, the verification of Step 2 amounts to show that given an affine horseshoe with large fractal dimension, we can perturb (in topology) the dynamics so that the projections along of Markov rectangles defining overlap considerably. Of course, since has large fractal dimension, the Markov rectangles have large size. Thus, if we could perturb independently these Markov rectangles, then the desired overlap property would follow. Of course, it is not possible to perturb independently nearby Markov rectangles, but an ingenious probabilistic argument of Moreira-Yoccoz shows that there exists a large (almost full) probability that a random perturbation of the Markov rectangles will move “independently” Markov rectangles reasonably far away from each other.

Finally, let us quickly discuss Step 1. The basic idea is very simple. As we mentioned above, the Lyapunov exponents of all measures supported on are -close to constants (namely, ), that is, we are almost affine. Using this information, one selects a tiny grid of our space, and one linearizes (after -perturbation) the dynamics inside the elements of the grid. Of course, there is a boundary effect to control (in order to ensure that the dynamics at the boundary of the grid is well-defined) and, in this direction, Avila-Crovisier-Wilkinson use a “reversed doubling property” of the maximal entropy measure of (saying that the ratio of concentric balls are controlled in terms of the ratio of their radii: for some ) to show that this boundary effect is negligible.

For this sake, let us recall that, in the previous post of this series, we considered an arbitrarily fixed level of the systole function and we introduced a subset consisting of such that all its non-vertical saddle-connections have length . Then, we defined the set

and we studied the -measure of the subsets

of translation surfaces with systole that are “accessible” by and movements from . In this setting, the results of the previous post of this series can be summarized as follows:

• , where is a finite measure on ;
• for all ;
• is a density measure in the sense that

  

From this point, we will divide this final post into two sections. In the first one, we will formalize the idea that occupies most of for adequate choices of , so that the proof of Eskin-Kontsevich-Zorich regularity conjecture will be reduced to the computation of the -measure of . Then, in the final section, we will show that the set of such that has small -measure for large because these ‘s have a pair of non-parallel saddle-connections with a small angle (and is ).

1. A “perfect cancellation” result

Let . From the fact that imply that the level sets of the systole function have zero -measure, and, thus, is a continuous non-increasing function.

Furthermore, the fact that is a density measure implies that has a left derivative

equal to .

By combining this with the facts that and , we get that

Moreover, the Siegel-Veech formula says that for all , so that the previous estimate imply that

In particular, the left derivative is bounded.

From this, it follows that is absolutely continuous: indeed, if the left derivative of a continuous function is bounded by on an interval , then because, for any , one can check that the supremum of the such that is (in view of the assumption that for all ).

At this point, we are ready to “improve” the Siegel-Veech formula by showing that there exists a constant such that

Indeed, let

Since is absolutely continuous, it is the integral of its left-derivative. It follows that

for all , and, hence,

On the other hand, from (1), we know that

for all . Therefore, by taking , we obtain that

In summary, we showed that

Once we know that , we can show the following “perfect cancellation” equality:

In fact, it is obvious that , and, for the converse inequality, we can use that is the integral of its left derivative:

We call the equality

a “perfect cancellation” by the following reason. Recall that (where and ). So, by taking a sequence such that as , and by considering the sets , we conclude that

Since , we conclude that occupies most of in the sense that

In particular, our task of showing that

is reduced to show that, for each , and all sufficiently large (depending on ),

The proof of this last assertion will occupy the next (last) section of this post.

2. End of proof of EKZ regularity conjecture

Let us fix once and for all one of the ‘s, , as above, and let us consider the set . By definition, they consist of of the form with and possessing two non-parallel saddle-connections of lengths .

Without loss of generality, we can assume that the lengths of these two saddle-connections are comparable within a large multiplicative factor . Indeed, if is very large and the ratio of these saddle-connections is larger than , then the systole of is actually and, by the Siegel-Veech formula, the -measure of the set

is for large enough.

Next, we consider the set of giving birth to with a pair of saddle-connections of lengths comparable within a large multiplicative factor from the systole.

We affirm that, for large enough, the angle is small, that is, for each and , there are and such that if with and , then:

• either
• or has a pair of non-parallel saddle-connections of lengths making an angle .

Assuming momentarily this claim, let us complete the proof of EKZ regularity conjecture. In this direction, let us consider the first item above and let us show that the -measure of this event is if small enough. Recall that . Therefore, given , for all , the -measure of the subset of translation surfaces in with is

Since this quantity is for small enough, it suffices to estimate the -measure of the set of with where denotes the event described in the second item of our claim. Here, we observe that, by discreteness of the set of holonomy vectors of saddle-connections, is for small enough. Therefore, the -measure of the set of with is

so that the proof of EKZ regularity conjecture is complete.

Now, let us prove our affirmation. Since , has a saddle-connection in the direction of . Now, let us try to figure out what non-vertical vectors can represent (the holonomy vector of) a saddle-connection on such that the saddle-connection of represented by has length comparable to within the multiplicative factor .

Suppose that the first item of our claim is violated, i.e., . In this situation, as we saw in Subsection “On the action of the diagonal subgroup ” of the second post of this series, given , there exists a constant such that if , then for all . It follows that, if , then

In other words, if we start with a saddle-connection of whose holonomy vector satisfies , then is not comparable to within the multiplicative factor . Equivalently, if is the holonomy vector of a saddle-connections of leading to saddle-connections of whose lengths are between and , then .

At this point, our task consists into checking that if with is the holonomy vector of a saddle-connections of leading to saddle-connections of whose lengths are between and , then the angle between and is if is large enough and . However, this last property is not hard to get: if and the angle between and is , then

for some . Since expands horizontal vectors, one can check that

for and large enough depending on , e.g., and . In other terms, if with , then a saddle-connection of () of length non-parallel to a length-minimizing saddle-connection of must come from a saddle-connection of of length making an angle with the vertical direction.

In summary, we showed that if the first item of our claim is violated, then the second item holds. This proves the claim and, a fortiori, the proof of EKZ regularity conjecture.

Let be a certain (fixed) level of the systole function on a connected component of a stratum of the moduli space of translation surfaces of genus . Denote by the set of such that all non-vertical saddle-connections have length and, for each , consider the set of translation surfaces with systole having the form for some , and . In other words, using the notation introduced in the previous post,

Geometrically, consists of the pieces of arcs of hyperbola below the threshold in the figure below:

In this notation, given a -invariant probability measure on , we want to compute the measure of in terms of , that is, we want to determine how “fat” is the set of translation surfaces that are “accessible” (via and movements) from .

In fact, a precise answer to this question will occupy this entire post and, in the next (and last) post of this series, we will use this answer to estimate the -measure of the set (of translation surfaces with two non-parallel saddle-connections of lengths ) as follows. Firstly, we will see that captures “almost all” translation surfaces in (for , i.e., , conveniently chosen). In particular, we will reduce the problem of measuring to the task of estimating . Then, we will see that the translation surfaces generating a translation surface have a pair of saddle-connections with a very small angle (with tending to zero as ) and it is not hard to see that this angle condition corresponds to a subset of with arbitrarily small “density” in .

We organize this post into two sections. In the first section we will explain how the variant of Rokhlin disintegration theorem from the previous post of this series can be used to describe the -measure of in terms of the measure of with respect to a certain obtained after disintegrating along certain pieces of -orbits. Then, in the second section, we will explain why the measure obtained from this disintegration process is a sort of “density” measure.

1. Construction of the “density measure” on

Let be a -invariant probability measure on a connected component of a stratum of the moduli space of translation surfaces of genus . Given such that , consider the set

of translation surfaces with systole accessible from from and movements, where is the set of translation surfaces with systole such that all non-vertical saddle-connections have length .

Note that we can write

where the symbol denotes a disjoint union: indeed, if with , then ; since and have systole and , it follows that ; in particular, and, from the definition of , we also deduce that .

Next, we observe that the infinitesimal generator of is tangent to . Indeed, since fixes the vertical direction while changing the horizontal direction only a little bit if is close to zero, we have that for all and sufficiently close to zero.

Remark 1 Here, we implicitly used that is a manifold when talking about tangent vectors to . While this is true if the translation surfaces in have no non-trivial automorphisms (symmetries), in general is an orbifold but not a manifold. However, this little technical problem is easy to overcome: one can either take finite covers of that are manifolds (e.g., one can mark horizontal separatrices of the translation surfaces to “kill” automorphisms), or one can write as a finite union of -invariant manifolds (as in our preprint with Artur and Jean-Christophe). In any event, for the sake of this post, we will pretend that is a manifold.

In particular, since is a codimension submanifold of (we are fixing the value of the systole, a codimension 1 condition, and the direction of the lenght-minimizing saddle-connections, another codimension 1 condition), we can select near any point a codimension submanifold locally transverse to the infinitesimal generator of . Using these ‘s, we can represent as

with , close to , and . At this point, we are ready to use the technology of the previous post to understand the measure on in terms of a density measure on . Indeed, we begin by noticing that has positive -measure. In fact, by Fubini’s theorem and the -invariance of , we have that

where is any probability measure on . By taking a “normalized” version of Haar measure of (say multiplied by an adequate everywhere positive density in order to get a probability measure) and by noticing that, for each with , the set of elements has non-empty interior in and thus positive -measure, we deduce that is positive as soon as the -measure of the set is positive.

Once we know that , we can apply the variant of Rokhlin’s theorem in the previous post to the to deduce that locally

where and is a finite measure on .

Next, we recall from the Subsection “The decomposition and Haar measure on ” of the previous post that . Therefore, we can write

where is a finite measure on .

From this formula, we can answer the question at the beginning of this post, i.e., we can compute the -measure of the set

In fact, by (1)

where and is the length of the interval . Now, we recall that the quantity was computed in Subsection “On the action of the diagonal subgroup ” of the previous post:

where . By plugging this into the integral expression for above, we get that:

Finally, the integral on the right hand-side was computed in the Subsection “The decomposition and Haar measure on ” of the previous post and its value is . In summary,

Proposition 1

In other terms, the -measure of the subset of captured by is an explicit (and simple) function of the quantity where was obtained from after disintegration along pieces of -orbits.

Intuitively, we want to think of as a sort of “density measure” on and we wish to say that occupies most of the level of the systole function, but these properties are not automatic from Rokhlin’s disintegration theorem. On the other hand, as we are going to explain in the next section, we dispose of all elements to give a precise meaning for the intuition that is a density measure.

2. Why is a density measure?

The main result of this section is:

Proposition 2 For each , denote by the slice

Then,

Of course, this proposition says that is a density measure in the sense that can be recovered from the -measures of slices near the level set of the systole function.

We will divide the proof of this proposition into two parts. Firstly, we will construct a regular part of the slice consisting of translation surfaces obtained by applying a “twisted Teichmüller flow” to and we will show that this regular part has -measure ; in particular, this computation of the -measure of the regular part of shows that . Secondly, we will show that the -measure of the singular part of the slice , i.e., the complement of the regular part, is and this will complete the proof of the desired proposition.

In order to organize the discussion of Proposition 2, we will treat the regular and singular parts of separately in the following two subsections below.

2.1. Regular part of slices

Let and consider the following “twisted Teichmüller flow”:

when . Note that is injective and has systole .

In this language, the regular part of the slice is:

For the computation of the -measure of , it is convenient to study the following measure on :

for any Borel set .

Proposition 3 doesn’t depend on : in fact, on .

Proof: As we saw in the previous section, if we write as with .

On the other hand, by definiton, if with , then

So, we will be able to compute if we can express in terms of the decomposition, i.e.,

Note that, for small enough, the vector satisfies and . By the results in Subsection “The decomposition and the Haar measure of ” of the previous post, it follows that has a decomposition . Furthermore, by a direct computation similar to the one in Subsection “The decomposition and the Haar measure of ” of the previous post for the computation of the density of Haar measure in -coordinates, one can check that, for close to zero, , and , so that

for any Borel set . It follows that as desired.

Corollary 4 The -measure of the regular part of the slice is:

In particular,

2.2. Singular part of slices

The main proposition of this subsection is:

Proposition 5 The singular part of the slice has -measure

as .

Observe that, by definition, the singular part of is contained in the set of translation surfaces with a saddle-connection of length non-parallel to a length-minimizing one. For each , let denote the minimal angle between two non-parallel saddle-connections of length .

The proof of Proposition 5 follows from the following two lemmas:

Lemma 6 (“small angles”) Given , there exists such that

Lemma 7 (“big angles”) For each ,

Let us start with the case of big angles. By Fubini’s theorem and the -invariance of , we have that

for any Borel set , where is a Haar measure on . By applying this formula with , our task is reduced to estimate the quantity

for each . By definition, if and , then has systole (for small enough) and has a pair of non-parallel saddle-connections and of lengths and angle such that and have lengths between and . In other words, using the notations from Subsection “Euclidean norms under -action on ” from the previous post,

where and are non-parallel holonomy vectors of saddle-connections of with lengths between and and making angle . From the main result of the Subsection “Euclidean norms under -action on ” from the previous post, the Haar measures of each of the sets are where the implied constant depend only on , i.e., . Thus, the proof of Lemma 7 will be complete once we dispose of upper bounds on the number of pairs as above and, as a matter of fact, these bounds exist in the literature: indeed, among several other things, H. Masur (see also Theorem 5.4 of this paper of A. Eskin and H. Masur) showed that the number of saddle-connections of length of a translation surface with systole is uniformly bounded by a constant depending only on .

Now, let us deal with the case of small angles. Given , we can select small enough so that the -measure of the set of translation surfaces with two non-parallel saddle-connections with angle and lengths between and is . Next, we select small enough so that

for all . In particular, if we consider the subset of consisting of such that a length-minimizing saddle-connection makes angle with the vertical, then, from the fact that , one can check that the sets are mutually disjoint for . By combining this fact with the -invariance of , we deduce that

Since , we conclude that

Finally, since differs from only by the extra condition that a length-minimizing saddle-connection makes angle with the vertical, we can use the -invariance of to obtain that

Of course, this completes the proof of Lemma 6.

At this point, we can summarize the discussion of this post as follows. We wrote as and we used this to show that the -measure of the set of translation surfaces accessible from by and movements is

where is a “density” measure in the sense that

with denoting the slice .

Next time, we will use this information and the Siegel-Veech formula to complete our solution of Eskin-Kontsevich-Zorich regularity conjecture.

1. Some facts about

In this section, we’ll discuss 3 results where:

• we’ll study the action of on the Euclidean norms of non-collinear vectors (cf. Proposition 2 below),
• we’ll compute the Haar measure on a certain open subset of (cf. Proposition 3 below), and
• we’ll estimate the amount of time that the diagonal subgroup keeps the Euclidean norm of the vector below a given threshold (cf. Proposition 4 below).

During this entire section (and, actually, in the other posts of this series also), we will use the following notation:

• is the standard basis of and denotes the Euclidean norm of ;
• is the (positive) diagonal subgroup of ;
• is the rotation by angle ;
• is a lower triangular matrix.
• is a upper triangular matrix.

1.1. Euclidean norms under -action on

The following lemma computing the Haar measure on is well-known, see, e.g., A. Knapp’s book (especially Chapter 5 on Iwasawa decomposition and integral formulas, and Equation (10.7) in Chapter 10).

Lemma 1 The map from to is a diffeomorphism and the measure is sent to a Haar measure by this diffeomorphism.

Using this lemma, we will estimate the Haar measure of the set of elements of with bounded norm keeping two fixed non-collinear vectors inside a given annulus.

Proposition 2 Let be vectors with . Then, the Haar measure of the set

is when is small. Here, the implied constant is uniform when is bounded away from .

Remark 1 For the application we have in mind, it suffices to know that the Haar measure of is , but we prefer to state this proposition in this form in order to highlight the following phenomenon. A naive computation reveals that the Haar measure of the set of elements of with bounded norm keeping a given vector inside the annulus is . In particular, it would be natural to think that has Haar measure as we are trying to confine two non-collinear vectors inside the annulus . However, as we’ll see during the proof of the proposition, this intuitive picture is wrong because of certain singularities (critical points) of “square root” type develop (in other words, the events of confining two non-collinear vectors inside an annulus are not always independent). Fortunately, the singularity is mild and the estimate of the Haar measure of is sufficient for our future purposes.

Proof: The set is empty (for small ) unless , so we’ll assume that and satisfy these inequalities. Also, from the right-invariance of the Haar measure under , we can suppose that with .

Now, let us write and , and let us introduce the function

Clearly doesn’t depend on , so that we will write in the sequel.

We have . Thus, the Jacobian matrix of is

Since and , this matrix is invertible unless , i.e., and are collinear () or orthogonal ().

We will complete the proof of this proposition by considering the following cases:

• Suppose that with and . Then, with when . Since , we can’t have

  

  at the same time if is small enough depending only on , i.e., . In other words, in this situation, the set is empty for small enough depending on .

• Suppose that and are orthogonal, i.e., . Then, . The condition determines an interval of ‘s of length : indeed, is equivalent to ; since , the interval has length . Next, we observe that, for a fixed in this interval, the condition determines an interval of ‘s of length (and exactly of order in the worst case ): in fact, is equivalent to , that is, ; since (as we mentioned above) and , we get from that , and, thus, belongs to an interval of size . From Lemma 1, it follows that, in this situation, the Haar measure of is . Moreover, the same estimate is true if we relax the condition to in the definition of .
• Assume that there exists such that and is orthogonal to . In this case, the fact that has Haar measure follows from the previous item after performing a right-translation of by .
• Assume that none of the assumptions hold. Then, the Jacobian matrix of is invertible on the set and the norm of its inverse is uniformly bounded by a constant depending only on how far is from . In this case, from the inverse function theorem and Lemma 1, we deduce that the Haar measure of is where the implied constant depends only on .

This completes the proof of the proposition.

1.2. The decomposition and Haar measure on

Let .

Proposition 3 The map from is a diffeomorphism onto . Furthermore, up to a multiplicative constant, the restriction to of a Haar measure is equal to in the coordinates .

Proof: The vertical basis vector is fixed by and it is mapped to under . Therefore, for , , , the matrix satisfies and , i.e., belongs to .

Conversely, given a vector with and , there exists a unique pair depending smoothly on with . Indeed, this follows from the facts that the matrix moves vectors along hyperbolas and, for , the matrix moves along the arc of unit circle containing between the hyperbolas and , see the figure below:

This proves the first assertion of the proposition because, after writing a vector with and uniquely as for , , if we are given a vector such that , then we can write by choosing the unique such that where is the horizontal basis vector of .

Now, let us write the restriction of a Haar measure to as where is a positive smooth density function. Since any Haar measure on is left and right invariant (i.e., is unimodular), we deduce that depends only on , i.e., . In order to compute , we will use the left-invariance of Haar measures under . More concretely, let us fix , and let us consider tiny open sets around the origin and their respective images under . In terms of matrices, this amounts to consider the equation:

for close to , and , , . For sake of simplicity, since is fixed, we will omit the dependence of the functions on in what follows. In this language, we have from the -invariance of Haar measures and the change of variables formula that where is the determinant of the Jacobian matrix at the origin . In particular, our task is reduced to show that . Keeping this goal in mind, let us notice that implies that , and in Equation (1). Thus, we have that

and

In other words, the Jacobian matrix has the form

at the origin , so that . Hence, it remains only to show that . In this direction, we apply both matrices in Equation (1) to the vertical basis vector to get the equations:

and

By multiplying these equations, we get the relation:

By taking the partial derivative with respect to , we deduce that

Therefore, since at the origin , we have that

that is,

Now, we can plug this information into Equation (3) to calculate . Indeed, by taking the partial derivative with respect to in (3), we get

Since at the origin , we obtain that

By combining this relation with (4) we conclude that

Of course, this proves the second assertion of the proposition.

Remark 2 Note that the computations above give all entries of the Jacobian matrix at the origin but . Evidently, the knowledge of this particular entry was irrelevant for the computation of in the previous proposition, but the curious reader is invited to compute this entry along the following lines. By applying both matrices in (1) to the horizontal basis vector , one gets two relations:

and

By taking the partial derivative of the second relation above with respect to at the origin and by plugging the values and already computed, one deduces that

i.e.,

1.3. On the action of the diagonal subgroup

Let be a given rotation. For , define:

Geometrically, is the interval of times such that, after applying to the unit vector , the norm of the vector stays below the “threshold” .

Proposition 4 The set is empty if and only if . If , then, by writing with , one has that is an open interval of length .

Proof: By definition, if and only if , i.e.,

By performing the change of variables or depending on whether or , and by multiplying the previous inequality by , we get or . This inequality has a solution if and only if the discriminant of the corresponding second degree equation is positive, i.e.,

This proves the first assertion of the proposition.

If and we write with , then , so that . It follows that the solutions of

belongs to the open interval between the roots

of . By recalling the change of variables , we deduce that is an open interval where . In particular, the length of is

This completes the proof of the second assertion of the proposition.

Later in our discussion, we will combine Propositions 3 and 4 to derive some measure estimates of the sets of translation surfaces with short saddle-connections, and, for this sake, we will need the following equality:

Lemma 5

Proof: The change of variables gives

Since (as ) and for (by integration by parts), we deduce that

By Leibniz formula, we know that . By plugging this into Equations (6) and (5), we obtain the desired lemma.

Also for later use, we will need the following corollary of the proof of Proposition 4.

Corollary 6 Given , there exists a constant such that, if

then

for all .

Proof: Recall that , so that

In particular, from the definition of , we have that for . By combining these two estimates, we deduce that

On the other hand, from the proof of Proposition 4, we know that, by writing ,

for every .

By hypothesis, (i.e., ), so that we conclude from the previous inequality that

By plugging this into our estimate of above, we deduce that

and thus the proof of the corollary is complete.

At this point we have all elementary facts about and its action on , and we will close this post with the following remark:

Remark 3 Historically, the proof of Proposition 4 was the starting point of our solution with A. Avila and J.-C. Yoccoz of the Eskin-Kontsevich-Zorich regularity conjecture. Interestingly enough, after Artur, Jean-Christophe and I completed our article, A. Eskin pointed out to us that this lemma was previously known by G. Margulis and, indeed, they planned to include this fact in one of their joint articles but they ended up by not relying on it.

2. Rokhlin’s disintegration theorem

Very roughly speaking, given a probability measure on a space and a partition of , Rokhlin’s disintegration theorem concerns the problem of writing/decomposing as a “superposition” of (conditional) probability measures supported on the elements of the given partition. In other words, Rokhlin’s theorem addresses the question of disintegrating into supported on

Of course, such a decomposition might not exist in general, but Rokhlin’s disintegration theorem provides fairly general conditions ensuring the existence and uniqueness of disintegrations/conditional measures.

Theorem 7 (Rokhlin’s disintegration theorem) Let be a Lebesgue space and let be a measurable partition of , i.e., there is a sequence of finite partitions of with the following properties:

• denoting by denotes the element of a given partition containing , one has that for all and ;
• the sequence is monotonic: for all and , ;
• is the limit of : for all , .

Then, there exists a system of conditional measures for , i.e.,

• for all , (i.e., is supported on );
• for all , (i.e., is constant on elements of the partition );
• for any , the function is measurable and

  

  (i.e., disintegrates ).

Moreover, the system of conditional measures is essentially unique: if is another system of conditional measures, then for -almost every .

Remark 4 In some sense, Rokhlin’s disintegration theorem is a sort of martingale convergence theorem: indeed, for the finite partitions , it is easy to disintegrate by letting be the normalized restriction of to , and one has to work to show that the desired ‘s are the limits of ‘s. This point of view is nicely explained in these notes of M. Viana here (where a proof of a particular case of Rokhlin’s disintegration theorem is proved).

For our future applications, the probability measure will leave on a space with a partition given by pieces of a natural action of a Lie group and will be “invariant” under this Lie group. In this context, Rokhlin’s disintegration theorem alone will not suffice for our purpose because we will want to know that the conditional measures are essentially pieces of Haar measures. Here, it is worth to point out that similar settings were already considered by other authors, but the exact statement (below) needed in our applications in the next posts of this series doesn’t seem to appear in the literature. So, we will close today’s post with the following variant of Rokhlin’s disintegration theorem.

Let be a Polish space and let be a Lie group. Denote by a left-invariant Haar measure on and a left-invariant metric on .

Let act on by and denote by the canonical projection. Let be an open subset and let be a probability measure on .

Note that comes with a natural measurable partition . Thus, by Rokhlin’s disintegration theorem, has a system of conditional measures that we can think as a probability measure on the fiber of over .

We will assume that is invariant, that is, for all measurable and all such that , we have . In this context, we can assert that the conditional measures are pieces of Haar (-)measure on .

Proposition 8 If is invariant, then, for -almost every , has finite (Haar) -measure and

Proof: For -a.e. , the conditional measure is “invariant”. Intuitively, this follows from the uniqueness of the system of conditional measures and the invariance of , but one has to be take a little bit of care about the meaning of “invariant” for .

For our purposes, we will consider the following notion. Let be a countable dense subset of and let be the countable set of balls of the left-invariant metric on centered at points in with positive rational radius.

Since and are countable, we can use the uniqueness of conditional measures and the invariance of (plus the fact that countable unions of sets of zero measure have zero measure) to deduce that, for -almost every , if and satisfy and , then , i.e., is “invariant” if we test the invariance properties exclusively with elements and of the countable (dense) sets and .

In any case, the fact that and are dense are enough to deduce that ‘s are Haar measures from the “weak” form of invariance. Indeed, notice that it suffices to show that ‘s are absolutely continuous with respect to because the “weak” invariance property is good enough to ensure that the density is constant. Now, once we reduced our task to the absolutely continuity of , we can proceed as follows. Given , with radius , by the nice (homogeneity) properties of the Haar measure on a Lie group, we can find an integer (with independent of ) and some elements such that the balls are mutually disjoint and contained in . By the “weak” invariance of , we obtain that

so that , and, hence, is absolutely continuous with respect to .

By the end of the conference, A. Zorich mentioned a conjecture in his groundbreaking article with A. Eskin and M. Kontsevich concerning the regularity of ergodic -invariant probability measures on moduli spaces of Abelian differentials.

Right after the end of the conference, Jean-Christophe Yoccoz, Artur Avila and I discussed the possibility of using “soft” approaches to this conjecture in the sense that we wished to stick to “elementary” properties of , but not on specific features of translation surfaces.

After a couple of further discussions, mostly in Paris (at Collège de France) and Rio (at IMPA during the first Palis-Balzan conference), we managed to reunite in this preprint here the “soft” elements about and its actions on and moduli spaces leading to a solution of the regularity conjecture of A. Eskin, M. Kontsevich and A. Zorich.

In a series of four posts, we’ll explain the regularity conjecture of Eskin-Kontsevich-Zorich and the “soft” methods in the preprint by A. Avila, J.-C. Yoccoz and myself solving this conjecture.

More precisely, we’ll discuss in today’s post the statement (and motivations) of Eskin-Kontsevich-Zorich’s regularity conjecture and we’ll describe the general lines of our solution of this conjecture. Then, in the next post of the series, we’ll explain the first step in our solution, namely, the proof of 3 elementary facts about and its action on and some classical results about conditional measures. After this, in the third and fourth posts of the series, we’ll give an answer to Eskin-Kontsevich-Zorich’s regularity conjecture by using the results on conditional measures to “transfer” the elementary results about the action of on to the moduli spaces of translation surfaces.

Closing this introduction, let us stress out that, while the Eskin-Kontsevich-Zorich conjecture concerns -invariant probabilities in moduli spaces of Abelian differentials, the next post of this series will concern only , its action on and conditional measures, and thus it might be of independent interest. In particular, it is not necessary to have prior knowledge of Abelian differentials and translation surfaces to read the second post of this series.

1. Eskin-Kontsevich-Zorich regularity conjecture

The dynamics of the natural action on the moduli space of Abelian differentials is a fascinating topic related to the renormalization of interval exchange transformations, translation flows and billiards in rational polygons. In fact, this topic was already discussed a couple of times in this blog (see, e.g., these posts here for basic definitions, motivations and applications), and, for the sake of this section, we will assume that the reader is familiar with Abelian differentials as translation surfaces (that is, compact surfaces obtained by gluing by translations the parallel sides of a finite collection of polygons in the plane) and the dynamics of the Teichmüller geodesic flow, and the Kontsevich-Zorich cocycle.

In their groundbreaking article mentioned above, A. Eskin, M. Kontsevich and A. Zorich showed the following formula for the sums of non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle with respect to an ergodic -invariant probability measure on the moduli spaces of unit area translation surfaces of genus :

Here, is a combinatorial term depending only on the orders of the zeroes of the Abelian differentials in the support of , and is a so-called Siegel-Veech constant related the geometrical problem of counting cylinders in the translation surfaces in the support of .

In the future we will come back to this formula in more details, but for now let us just point out that, very roughly speaking, this formula is derived as follows.

Firstly, the sums of Lyapunov exponents are related to the curvature of the determinant line bundle of the Hodge bundle over the moduli space of translation surfaces by the so-called Kontsevich-Forni formula. In special cases this formula of Kontsevich and Forni is already sufficient to compute Lyapunov exponents, but in general it is not easy to calculate the curvature of the determinant of the Hodge bundle. At this point, A. Eskin, M. Kontsevich and A. Zorich use an analytic version of the Grothendieck-Hizerbruch-Riemann-Roch formula to convert the Kontsevich-Forni formula into an equality of the form:

where is an integral expression involving the logarithm of the determinant of the flat Laplacian of translation surfaces in the support of . In some sense, the appearance of the combinatorial term (or at least the factor ) is “natural” if one recalls (or compares) with the Noether formula (a version of Grothendieck-Hizerbruch-Riemann-Roch formula for surfaces).

If the moduli spaces of translation surfaces were compact, an integration by parts argument would say that . However, it is well-known that the moduli spaces of translation surfaces are not compact, and thus we get a boundary contribution making that is not trivial in general.

At this stage, the idea of A. Eskin, M. Kontsevich and A. Zorich is very simple: by carefully performing the integration by parts argument, one can relate , an integral on the moduli space, to a Siegel-Veech constant , a geometric quantity related to the flat geometry of translation surfaces in the support of , if satisfies a certain technical condition called regularity allowing to treat some “boring” terms in the integration by parts as “error” terms.

Formally speaking, the regularity condition is defined as follows.

Given a translation surface , recall that a (maximal, flat) cylinder is a maximal collection of parallel closed regular geodesics of . The height of a cylinder of is the distance across , the circumference of is the length of its waist curve and the modulus is .

In the picture below, we describe a L-shaped square-tiled surface with two horizontal cylinders , , with waist curves . The circumference of , resp. is , resp. , and the height of , , is . In particular, and .

Using this notation, given and , let be the set of translation surfaces in the support of possessing two non-parallel cylinders and with moduli and for .

We say that is regular when there exists such that , i.e., .

As it turns out, all known examples of ergodic -invariant probability measures are regular: for example, the regularity of the so-called Masur-Veech measures was shown by H. Masur and J. Smillie, and A. Eskin, H. Masur and A. Zorich (see also this recent preprint of D.-M. Nguyen for some related results). For this reason, A. Eskin, M. Kontsevich and A. Zorich conjectured that all ergodic -invariant probability measures are regular.

In other words, the Eskin-Kontsevich-Zorich formula is stated for regular ergodic -invariant probability measures in their paper simply because at the very last step they need this condition to justify a certain integration by parts argument, but they believe that their formula holds for any ergodic -invariant probability measure.

In our preprint, A. Avila, J.-C. Yoccoz and I confirmed the Eskin-Kontsevich-Zorich’s regularity conjecture by showing the following slightly stronger result.

Recall that a saddle-connection in a translation surface is a compact geodesic segment joining singularities (zeroes) of such that has no singularity (zero) of in its interior. In the picture below, we depicted a -shaped square-tiled surface and we marked (in blue) four saddle-connections.

Let denote the set of translation surfaces in the support of a -invariant probability measure possessing two non-parallel saddle-connections of length .

Theorem 1 (A. Avila, C.M. and J.-C. Yoccoz) One has that .

This result is slightly stronger than the regularity conjecture because the boundaries of cylinders are unions of saddle-connections and, a fortiori, .

In order to put Theorem 1 in perspective, let us mention that the set of translation surfaces with some saddle-connection of length has measure

In particular, our Theorem 1 says that occupies a small proportion of .

The proof of the estimate is due to W. Veech, and A. Eskin and H. Masur, and it is based on the so-called Siegel-Veech formula. Actually, as it is pointed out by A. Eskin and H. Masur in their article, the Siegel-Veech formula has very little to do with moduli spaces and it is essentially something about the action of on . So, let us close this section by explaining (quickly) how this formula works.

Let be a space where acts by preserving a probability measure , and consider a function associating to each a set with multiplicity (i.e., is a set of non-zero vectors of with weights). For our purposes, we will take as a moduli space of unit area translation surfaces with fixed combinatorial data, and, for each , the set is the (discrete) set of holonomy vectors of saddle-connections of .

We will impose the following conditions on :

• varies linearly with , i.e., for every and .
• for each , there exists a constant such that the cardinality of the intersection of with the ball of center and radius is ; moreover, the constant can be chosen uniformly on compact subsets of .
• there are and such that the function belongs to .

The (non-trivial) fact that these conditions — especially the second and third items above — hold for the particular case of a moduli space of translation surfaces and the (holonomy of) saddle-connections function was verified by A. Eskin and H. Masur in their article.

Coming back to the general setting, given a real-valued function of compact support on , let us define its Siegel-Veech transform as

In this language, the Siegel-Veech formula is

where is the so-called Siegel-Veech constant of (with respect to ). At first sight, the Siegel-Veech formula looks tricky to prove (as and are “arbitrary”), but, as it turns out, this formula becomes easy to derive if we notice that

is a non-negative linear functional on , that is, the integration of Siegel-Veech transforms induces a measure on : indeed, this linear functional is well-defined because is finite, bounded on compact sets and by our assumptions on . Furthermore, since varies linearly with , it is not hard to see that this measure on is -invariant. Since the linear combinations of the Dirac measure at the origin and the Lebesgue measure are the sole -invariant measures on , it follows that this measure has the form

Finally, since , it is possible to check that , so that the Siegel-Veech formula holds (with ).

Once we know the Siegel-Veech formula, we can deduce that by applying this formula with (a smooth “version” of) the characteristic function of the ball :

In any event, after this little digression, it is time to explain some key steps towards Theorem 1.

2. General lines of the proof of Theorem 1

The basic idea behind the proof of Theorem 1 is the following. In some sense, we will perform an “orbit by orbit estimate” (with respect to -action) saying that the Haar measures of the intersection of with certain pieces of -orbits are . Then, we will use a sort of conditional measure argument to put together these “orbit by orbit estimates” to get the global estimate for in Theorem 1.

A little bit more precisely, our strategy is the following. Given a translation surface , let denote the systole of , that is, the length of the shortest saddle-connection(s). Given , let . Inside the -level of the systole function , we consider the sets

and

where denotes the rotation by .

From the set , we can “access deeper levels” of the systole function via the set

Indeed, the choice of and is guided by the fact that the vector is shorter than the (unit) vector for , , so that the systole of is smaller than the systole of .

Furthermore, is an interesting way to access because it is not hard to check that the sets for and form a nice (measurable in the sense of Rokhlin) partition of . In particular, by the -invariance of , we will be able to compute the -measure of subsets of in terms of the Lebesgue measure on , the Lebesgue measure on the circle and a certain “density measure” on .

Using this disintegration, we can “transfer” mass from to deep levels , , as follows. Firstly, we will show that, for , there is an open interval of whose length is explicitly computable such that for all . Geometrically, the set of for , , correspond to the pieces of segments (“hyperbolas”) below the threshold .

Since the codimension 2 subsets are represented by points, the picture above is a simplified version of the following more “complete” picture:

In particular, by the disintegration results in the previous paragraph, we’ll be able to show that the -measure of is at least

At this point, the idea is very simple. We will show that there is a (positive) constant such that:

• as , the -measure of is , and
• there exists a sequence with as such that the densities are .

Intuitively, this says that the densities of are almost “maximal”. Indeed, at first sight the factor of “1/2” might seem strange, but, as we will show, in general, the density of is given by where . In particular, by L’Hôpital rule, we expect that

if .

In any case, putting these facts together, we deduce that

From this, we get that the set of translation surfaces with systole “accessed” from occupies most of in the sense that its complement has -measure for all .

Now, once we know that most translation surfaces with systole “come” from , we will complete the proof of Theorem 1 by showing that the translation surfaces leading to translation surfaces are (essentially) those with two non-parallel saddle-connections of lengths comparable to making a very small angle . Here, “very small angle” means that becomes close to zero for is sufficiently large (depending on ). Then, since the -density of the set of those is small, say , for small, i.e., large, we can use again that disintegrates as to conclude that the -measure of is for large, as desired.

Of course, there are plenty of details to check in this scheme and the next installments of this series will formalize the ideas in this section. In particular, in the second post we will make some elementary estimates on allowing to compute the length of the interval introduced above (among other things). Then, we’ll complete the second post with some facts about conditional measures that we’ll use in the third post to define (and study) the “density measures” on . Finally, the last post will serve to formalize the “transport of mass” scheme described in the last 6 paragraphs above.

In his talk, Sylvain presented two of the main results of his work with Avila and Wilkinson (see Theorem 2 and Theorem 3 below), and he sketched the proof of one of them (namely, Theorem 2). Then, after his talk, he told me that he plans to discuss the proof of the other main result next Friday (February 22, 2013) at Eliasson-Yoccoz seminar in Jussieu (University Paris 6 and 7).

So, I will proceed as follows: below I’ll reproduce my notes from Sylvain’s talk at LAGA, and, if I manage to take decent notes from Sylvain’s talk at Jussieu, then I’ll complete today’s discussion (in another post) by sketching the proof of the other main result of Avila-Crovisier-Wilkinson (namely, Theorem 3).

As usual, all mistakes/errors in this post are entirely my responsibility.

1. Setting and statement of main results

Let be a compact, connected manifold (without boundary), and fix a smooth volume probability measure.

Denote by the set of -diffeomorphisms of preserving .

Our discussion will be “guided” by the following question:

Problem. Is ergodic for “most” ?

Of course, this problem is motivated by the so-called Boltzmann’s ergodic hypothesis “predicting” that the “answer should be yes”.

For , it was shown by J. Oxtoby and S. Ulam that is ergodic for a -generic (residual in Baire category sense) conservative homeomorphism.

On the other hand, the celebrated KAM theory shows (in particular) that the analog of Oxtoby-Ulam in higher smoothness is false:

Theorem (Kolmogorov, Arnold, Moser, …, Herman) For , there exists an open set such that every possesses a family of invariant torii of codimension 1 whose union is a Cantor set of positive -volume and the dynamics of on each invariant torus of this family is -conjugated to a (irrational) rotation on the standard torus .

In particular, the dynamics of is not transitive nor ergodic with respect to .

Remark 1 Some regularity condition for the validity of this theorem is necessary: for example, S. Crovisier and C. Bonatti used their closing lemma for pseudo-orbits to show that the dynamics of a -generic (in Baire category) conservative diffeomorphism is transitive. However, the exact regularity threshold for the validity of Theorem 1 is not known (to the best of my knowledge).

Remark 2 For the sake of our discussion, it is worth to point out that the absence of ergodicity in the open set of “KAM examples” above is intimately related to the absence of hyperbolicity in the following sense: if , then (i.e., there is no future or past exponential growth of the dynamics along the orbit of ) because is (-conjugated to) a rotation on and preserves a Cantor set lamination transversely to . In other words, the non-ergodicity of KAM examples is “natural” because they have no non-zero Lyapunov exponents along the orbits in .

Concerning Remark 2 above, let us recall that, in general, the Oseledets theorem asserts that, for -a.e. , there is a decomposition into Oseledets subspaces

and a collection of numbers such that

whenever .

From the dynamical point of view, it is natural to distinguish three types of vectors in depending on whether they are exponentially contracted, not exponentially contracted nor expanded, and exponentially expanded, that is,

where , and . We will call , and the stable, central and unstable Oseledets subspaces (resp.).

In this language, we say that is non-uniformly hyperbolic (in the sense of Pesin’s theory) if for -a.e. .

Here, the nomenclature “non-uniform hyperbolicity” is justified by the fact that the conditions or on Lyapunov exponents provide only asymptotic information on contraction or expansion, so that the time one has to wait before actually getting contraction or expansion might depend heavily on .

For sake of comparison, let us recall that is called Anosov if it admits a global uniform hyperbolic structure, i.e.,

where and are -invariant subbundles such that there exists with the property that is (uniformly) contracted by and is (uniformly) contracted by .

This “uniformity” of Anosov diffeomorphisms was exploited by D. Anosov and Y. Sinai to show the following result:

Theorem 1 Let be an Anosov diffeomorphism. Then, is ergodic.

The basic mechanism behind this theorem is the so-called Hopf’s argument, and the smoothness requirement is imposed to ensure that Hopf’s argument works. In particular, it is not known whether Anosov-Sinai theorem holds for .

In any case, an immediate corollary of Anosov-Sinai theorem is the fact that contains a -open set of ergodic diffeomorphisms whenever contains Anosov diffeomorphisms: indeed, this follows from the -openness of the condition of existence of global uniform hyperbolic structure (see, e.g., Hasselblatt-Katok’s book for more details).

In summary, it is not known that ergodicity is -open among conservative Anosov diffeomorphisms, but, at least, it is -open among conservative Anosov diffeomorphisms. Of course, this scenario motivates the question: given that we don’t know concrete examples of -open sets of ergodic -diffeomorphisms, what can be said about the -genericity (in Baire sense) of ergodicity?

The next theorem by A. Avila, S. Crovisier and A. Wilkinson gives an answer to this question:

Theorem 2 (A. Avila, S. Crovisier and A. Wilkinson) There exists a residual (i.e., -dense) subset such that for any :

• (ZE) either all Lyapunov exponents of -a.e. vanish,
• (NUA) or is non-uniformly Anosov in the sense that
  • has a (global) dominated splitting, i.e., there is a decomposition into -invariant subbundles such that dominates , that is, there exists with for any , unitary vectors (“the largest expansion along is dominated by the weakest contraction in , but, a priori, neither is assumed to be contracted nor is assumed to be expanded”).
  • for -a.e. , the fibers of and of the dominated splitting coincide with the stable and unstable Oseledets subspaces, i.e., and ,

  and is ergodic.

Remark 3 The attentive reader will notice that there is no ergodicity claim in item (ZE) and this is not by chance: in fact, the issue of ergodicity in the context of (ZE) is open and it is an interesting problem to understand this question even for particular cases of (ZE) such as “KAM-like examples”.

Remark 4 By definition, a non-uniformly Anosov diffeomorphism is non-uniformly hyperbolic in the sense of Pesin’s theory. In other words, non-uniformly Anosov systems form a intermediate class of dynamical systems between Anosov diffeomorphisms and non-uniformly hyperbolic diffeomorphisms.

Remark 5 In terms of Lyapunov exponents, this theorem says that -generically there is no “intermediate behavior”: either all Lyapunov exponents are zero (like in KAM examples) or they are all non-zero (and the “Oseledets splitting” is dominated).

Remark 6 For , this theorem was previously known from the works of R. Mañé and J. Bochi, and, more recently, J. Rodriguez-Hertz showed this theorem for .

Remark 7 The distinction between the possibilities (ZE) (zero exponents) and (NUA) (non-uniformly Anosov) can be done in terms of the metric entropy of . More precisely, we claim that (ZE) occurs if and only if , and (NUA) occurs if and only if . Indeed, by Ruelle’s inequality, . Thus, if all Lyapunov exponents vanish, , and, if , then (some) Lyapunov exponents are non-zero. In order to complete the picture, we invoke Pesin’s formula saying that Ruelle’s inequality is an equality in favourable situations: usually Pesin’s formula is stated for -diffeomorphisms, but W. Sun and X. Tian showed that it is also true in our current context (of -generic conservative diffeomorphisms) to show that if then (generically) all Lyapunov exponents must vanish.

After getting some positive result (namely, Theorem 2) for the question of -genericity of ergodicity, it is natural to come back to the question of openness/stability of ergodicity.

Partly motivated by the situation in Anosov-Sinai theorem, we say that , , is -stably ergodic if all -close to is ergodic (with respect to ).

By definition and Anosov-Sinai theorem, all Anosov diffeomorphisms are -stably ergodic. Evidently, one can ask for more general classes of stably ergodic dynamical systems, and, after the works of C. Pugh and M. Shub, K. Burns and A. Wilkinson, and F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures, we know that -partially hyperbolic diffeomorphisms satisfying certain mild (bunching and essential accessibility) conditions are -stably ergodic. Here, is called partially hyperbolic if there is a (global) splitting

into -invariant subbundles such that the stable direction is uniformly contracted by , the unstable direction is uniformly expanded by (i.e., contracted by ) and the central direction is dominated by and dominates . In the sequel, the set of -partially hyperbolic (conservative) diffeomorphisms is denoted by .

Remark 8 Of course, there is no a priori reason to stop at partially hyperbolic diffeomorphisms: indeed, it makes sense to ask stable ergodicity for conservative diffeomorphisms with dominated splittings and for conservative diffeomorphisms without dominated splittings. In the former case, A. Tahzibi constructed examples of stably ergodic conservative diffeomorphisms with dominated splitting (but not partially hyperbolic), but the general situation of stable ergodicity for diffeomorphisms with dominated splitting is not completely understood. In the latter case, as it was pointed out by S. Crovisier, there is no hope for stable ergodicity: in fact, one can exploit the absence of dominated splittings and a “pasting lemma” by A. Arbieto and myself to (generically) contradict ergodicity by producing periodic orbits possessing some invariant neighborhoods.

As it turns out, for the class of -partially hyperbolic (conservative) diffeomorphisms, it was conjectured by C. Pugh and M. Shub that there is no need for mild conditions for the validity of stable ergodicity:

Conjecture (Pugh-Shub). For , there exists a -open -dense subset of consisting of -stably ergodic dynamical systems.

In the direction of Pugh-Shub conjecture, A. Avila, S. Crovisier and A. Wilkinson show the following result:

Theorem 3 (A. Avila, S. Crovisier, A. Wilkinson) For , the set of ergodic diffeomorphisms in contains a -open, -dense subset of .

Remark 9 This theorem doesn’t exactly solve Pugh-Shub conjecture because they claim -density of the set of ergodic diffeomorphisms in (instead of -density).

Remark 10 For partially hyperbolic diffeomorphisms whose central direction is -dimensional, this result was previously shown by C. Bonatti, M. Viana, A. Wilkinson and myself, and, for whose central direction is -dimensional, this result was previously shown by F. Rodriguez-Hertz, J. Rodriguez-Hertz, A. Tahzibi and R. Ures.

At this point, as we already mentioned above, Sylvain offered to explain some ideas behind the proof of Theorem 2 while postponing the proof of Theorem 3 for another occasion. In fact, after his talk at LAGA (Univ. Paris 13), Sylvain told me that he intends to sketch the proof of Theorem 3 during a talk next Friday (22 February 2013) at Eliasson-Yoccoz seminar in Jussieu (Univ. Paris 6 and 7). In particular, it is likely that I will write a follow-up to this post (hopefully by the end of February/beginning of March) explaining what I could understand from Sylvain’s talk next Friday.

Anyhow, the next (and last) section of this post contains some highlights on the arguments used in the proof of Theorem 2.

2. Some ideas in the proof of Theorem 2

Let us consider a -generic conservative diffeomorphism. Recall that, by Oseledets theorem, for -a.e. , we have a decomposition into the stable, central and unstable Oseledets subspaces.

Define

Note that are -invariant subsets of such that

The first important ingredient in the proof of Theorem 2 is the following semicontinuity result for due to J. Bochi and M. Viana:

Theorem 4 (Bochi-Viana) Given , we can find a -invariant subset such that admits a dominated decomposition

where has dimension , has dimension and .

In fact, the following corollary of this result makes it clear why we called this a “semicontinuity result”:

Corollary 5 (Bochi-Viana) For -close to , the -measure of the symmetric difference of and is small.

Next, let us notice that the statement of Theorem 2 essentially amounts to say that generically only the “extremal cases”

or occur.

In this direction, we will need the following results of A. Avila and J. Bochi:

Theorem 6 (Avila-Bochi) is dense and is ergodic, i.e., there are and such that (mod ).

By putting together the semicontinuity theorem of Bochi-Viana with this theorem of Avila-Bochi, one obtains that:

Corollary 7 If , then .

Proof: By Avila-Bochi theorem, there are such that is a dense subset of . By Bochi-Viana theorem, there is a dominated splitting (with and ) over all . By definition of dominated splitting, the rates of contraction/expansion along and are distinct, so that a (global) dominated decomposition is not compatible with the presence of a positive measure subset where all Lyapunov exponents vanish (or, more generally, are all equal). It follows that .

At this point, the proof of Theorem 2 is essentially complete if we can show that generically the others (with ) do not show up (i.e., they have zero -measure).

Before proceeding in this direction, let us make a little detour to briefly explain how one gets ergodicity in the statement of Avila-Bochi theorem. As we told right after the statement of Anosov-Sinai theorem, one usually needs -regularity to obtain ergodicity (at least if one plans to use Hopf-like arguments…). In the context of Avila-Bochi theorem, one can “pretend” that -generic conservative diffeomorphisms “behave” like conservative diffeomorphisms as far as the classical ergodicity arguments are concerned. More concretely, by a result of A. Avila, one can -approximate any by some (this is trickier than one might think at first sight: while it is easy to perturb to improve its smoothness [say by convolution], but it is not so simple to regularize keeping the conservativeness condition; from the point of view of PDEs, this amounts to solve the Jacobian equation where is …), and one can use some “semicontinuity arguments” (like Bochi-Viana theorem…) to claim that behaves like a -generic .

Anyhow, let us “pretend” that and let us consider . By some results of A. Katok (strongly based on Pesin’s theory), it is known that we can “approach” by horseshoes, that is, there are periodic points , , such that

where . Here: and are the usual stable and unstable manifolds of the (hyperbolic) periodic point ; and are the stable and unstable manifolds of provided by Pesin’s theory; finally, means that and meet transversely.

The sets work as a sort of “homoclinic class in the sense of Pesin”, and, as it was shown by F. Rodriguez-Hertz, J. Rodriguez-Hertz, A. Tahzibi and R. Ures, they are good enough to play with a (generalized) Hopf argument:

Theorem 8 (Rodriguez-Hertz, Rodriguez-Hertz, Tahzibi, Ures) For any hyperbolic periodic point , is ergodic (if is ).

So, the proof of Avila-Bochi theorem (that is ergodic) will be complete if we can “connect together” the several “homoclinic classes” in order to obtain a single ergodic piece. Here, one can use the connecting lemma for pseudo-orbits of C. Bonatti and S. Crovisier saying that, for a -generic , any pair of hyperbolic periodic points and are connected in the sense that either or .

Coming back to the main discussion (i.e., the completion of the proof of Theorem 2), let us explain how to get rid of the “other” ‘s, that is, let us show that generically one has when .

The basic idea to “destroy” is to delete some central (zero) Lyapunov exponents. More concretely, we consider a subset such that is very small and has dominated splitting where and . By definition, the subbundle contains all zero Lyapunov exponents, so that we will destroy (or at least , a large piece of ) if we can convert some exponents in into non-zero Lyapunov exponents. In this direction, Avila, Crovisier and Wilkinson prove the following theorem:

Theorem 9 Let and consider an open set. Denote by the maximal invariant set of , i.e., and suppose that has a dominated splitting such that

for -a.e. . Then, for any , there exists -close to such that

In other words, this theorem says that, if the sum of Lyapunov exponents in is non-positive, then we can perturb the dynamics to get that the sum of Lyapunov exponents in is negative except for a set of small -measure.

Remark 11 For the case , this a similar result was proved by M. Shub and A. Wilkinson (for a important class of partially hyperbolic diffeomorphisms), and A. Baraviera and C. Bonatti (in general).

Using Theorem 9, one can destroy with by induction: starting with with and as large as possible, one can apply Theorem 9 to convert into a subset of or another with and .

Completing this post, let us say a few words about the proof of Theorem 9. We take a small ball of radius . Consider a diffeomorphism such that is the identity outside , and is a small rotation of towards say (for a mental picture you can think of as the -axis, as the -axis, as the -axis in and as a small rotation in the -plane) in the ball with the same center of and radius .

If we consider the action of the derivative of on along the orbits of points starting at , we will see that is tilted towards at the first step, and, in the next steps, the component of on gets contracted while the component of on stays about the same size.

In summary, this perturbation permits to the (sum of) Lyapunov exponents in slightly more negative along the orbits passing through the ball . Of course, there is a trade in this perturbation: since is an interpolation between a small rotation and the identity, it turns out that the effect on Lyapunov exponents of orbits passing through is the opposite of the expected one. In the case , this is not a big problem: this “boundary effect” can be controlled by arguing that most orbits (with respect to the volume ) will not see (possibly after adjusting the ratio between the radius of and to get closer to …). However, since we don’t know the set in advance, it could be that the a large part of orbits in the maximal invariant set will feel the boundary effect. At this point, Avila-Crovisier-Wilkinson borrow the following idea of J. Bochi: instead of performing a small rotation in the ball , one can iterate this produce to kill the boundary effect as follows: starting with , one iterates by and one lands in a ball . Now, one divides into a certain number of small balls and one performs another round of small independent rotations on each of these balls, and one continues by induction (i.e., subdividing the iterate of each ball and performing independent rotations). Of course, we are hidden lots of details (for instance, how to choose the parameters for the division of balls and for the rotations), but we will not comment more on this point (leaving the curious reader to consult Subsection 2.3 of Bochi’s paper for a more precise description of this idea).