Posted by: matheuscmss | February 19, 2013

## Eskin-Kontsevich-Zorich regularity conjecture I: introduction

From October 20 to 24, 2011, the conference Dynamics and Geometry organized by H. de Thélin, T.-C. Dinh and C. Dupont took place at Institut Henri Poincaré. This conference was marked by 7 interesting mini-courses by N. Mok, N. Sibony, J.-P. Demailly, A. Zorich, Y. Benoist, Y.-T. Siu and Y. Pesin: the full program is available here.

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 ${SL(2,\mathbb{R})}$-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 ${SL(2,\mathbb{R})}$, 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 ${SL(2,\mathbb{R})}$ and its actions on ${\mathbb{R}^2}$ 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 ${SL(2,\mathbb{R})}$ and its action on ${\mathbb{R}^2}$ 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 ${SL(2,\mathbb{R})}$ on ${\mathbb{R}^2}$ to the moduli spaces of translation surfaces.

Closing this introduction, let us stress out that, while the Eskin-Kontsevich-Zorich conjecture concerns ${SL(2,\mathbb{R})}$-invariant probabilities in moduli spaces of Abelian differentials, the next post of this series will concern only ${SL(2,\mathbb{R})}$, its action on ${\mathbb{R}^2}$ 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 ${SL(2,\mathbb{R})}$ 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, ${SL(2,\mathbb{R})}$ 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 ${\lambda_1^{\mu},\dots, \lambda_g^{\mu}}$ of the Kontsevich-Zorich cocycle with respect to an ergodic ${SL(2,\mathbb{R})}$-invariant probability measure ${\mu}$ on the moduli spaces of unit area translation surfaces of genus ${g\geq 1}$:

${\lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\frac{1}{12}\sum\limits_{i=1}^{\sigma}\frac{k_i(k_i+2)}{k_i+1}+c(\mu)}$

Here, ${\frac{1}{12}\sum\limits_{i=1}^{\sigma}\frac{k_i(k_i+2)}{k_i+1}}$ is a combinatorial term depending only on the orders ${k_1,\dots, k_{\sigma}}$ of the zeroes of the Abelian differentials in the support of ${\mu}$, and ${c(\mu)}$ is a so-called Siegel-Veech constant related the geometrical problem of counting cylinders in the translation surfaces in the support of ${\mu}$.

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 KontsevichForni 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:

$\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\frac{1}{12}\sum\limits_{i=1}^{\sigma}\frac{k_i(k_i+2)}{k_i+1}+I$

where ${I}$ is an integral expression involving the logarithm of the determinant of the flat Laplacian of translation surfaces in the support of ${\mu}$. In some sense, the appearance of the combinatorial term ${\frac{1}{12}\sum\limits_{i=1}^{\sigma}\frac{k_i(k_i+2)}{k_i+1}}$ (or at least the factor ${1/12}$) 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 ${I=0}$. However, it is well-known that the moduli spaces of translation surfaces are not compact, and thus we get a boundary contribution making that ${I}$ 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 ${I}$, an integral on the moduli space, to a Siegel-Veech constant ${c(\mu)}$, a geometric quantity related to the flat geometry of translation surfaces in the support of ${\mu}$, if ${\mu}$ 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 ${M}$, recall that a (maximal, flat) cylinder ${C}$ is a maximal collection of parallel closed regular geodesics of ${M}$. The height ${h(C)}$ of a cylinder of ${C}$ is the distance across ${C}$, the circumference ${w(C)}$ of ${C}$ is the length of its waist curve and the modulus ${\textrm{mod}(C)}$ is ${\textrm{mod}(C)=h(C)/w(C)}$.

In the picture below, we describe a L-shaped square-tiled surface with two horizontal cylinders ${C_i}$, ${i=1,2}$, with waist curves ${\gamma_i}$. The circumference of ${C_1}$, resp. ${C_2}$ is ${2}$, resp. ${1}$, and the height of ${C_i}$, ${i=1,2}$, is ${1}$. In particular, ${\textrm{mod}(C_1)=1/2}$ and ${\textrm{mod}(C_2)=1}$.

Using this notation, given ${K>0}$ and ${\varepsilon>0}$, let ${\mathcal{C}(K,\varepsilon)}$ be the set of translation surfaces ${M}$ in the support ${\mathcal{C}=\textrm{supp}(\mu)}$ of ${\mu}$ possessing two non-parallel cylinders ${C_1}$ and ${C_2}$ with moduli ${\textrm{mod}(C_i)>K}$ and ${w(C_i)<\varepsilon}$ for ${i=1, 2}$.

We say that ${\mu}$ is regular when there exists ${K>0}$ such that ${\lim\limits_{\varepsilon\rightarrow0}\mu(\mathcal{C}(K,\varepsilon))/\varepsilon^2=0}$, i.e., ${\mu(\mathcal{C}(K,\varepsilon))=o(\varepsilon^2)}$.

As it turns out, all known examples of ergodic ${SL(2,\mathbb{R})}$-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 ${SL(2,\mathbb{R})}$-invariant probability measures are regular.

In other words, the Eskin-Kontsevich-Zorich formula is stated for regular ergodic ${SL(2,\mathbb{R})}$-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 ${SL(2,\mathbb{R})}$-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 ${\gamma}$ in a translation surface ${(M,\omega)}$ is a compact geodesic segment joining singularities (zeroes) of ${\omega}$ such that ${\gamma}$ has no singularity (zero) of ${\omega}$ in its interior. In the picture below, we depicted a ${L}$-shaped square-tiled surface and we marked (in blue) four saddle-connections.

Let ${\mathcal{C}_2(\rho)}$ denote the set of translation surfaces in the support ${\mathcal{C}}$ of a ${SL(2,\mathbb{R})}$-invariant probability measure ${\mu}$ possessing two non-parallel saddle-connections of length ${\leq \rho}$.

Theorem 1 (A. Avila, C.M. and J.-C. Yoccoz) One has that ${\mu(\mathcal{C}_2(\rho))=o(\rho^2)}$.

This result is slightly stronger than the regularity conjecture because the boundaries of cylinders are unions of saddle-connections and, a fortiori, ${\mathcal{C}(K,\varepsilon)\subset \mathcal{C}_2(\varepsilon)}$.

In order to put Theorem 1 in perspective, let us mention that the set ${\mathcal{C}_1(\rho)}$ of translation surfaces with some saddle-connection of length ${\leq\rho}$ has measure

$\displaystyle \mu(\mathcal{C}_1(\rho))=O(\rho^2)$

In particular, our Theorem 1 says that ${\mathcal{C}_2(\rho)}$ occupies a small proportion of ${\mathcal{C}_1(\rho)}$.

The proof of the estimate ${\mu(\mathcal{C}_1(\rho))=O(\rho^2)}$ 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 ${SL(2,\mathbb{R})}$ on ${\mathbb{R}^2}$. So, let us close this section by explaining (quickly) how this formula works.

Let ${X}$ be a space where ${SL(2,\mathbb{R})}$ acts by preserving a probability measure ${\mu}$, and consider a function ${V}$ associating to each ${x\in X}$ a set ${V(x)\subset\mathbb{R}^2-\{(0,0)\}}$ with multiplicity (i.e., ${V(x)}$ is a set of non-zero vectors of ${\mathbb{R}^2}$ with weights). For our purposes, we will take ${X}$ as a moduli space of unit area translation surfaces with fixed combinatorial data, and, for each ${x\in X}$, the set ${V(x)}$ is the (discrete) set of holonomy vectors of saddle-connections of ${x}$.

We will impose the following conditions on ${V}$:

• ${V}$ varies linearly with ${SL(2,\mathbb{R})}$, i.e., ${g(V(x))=V(g(x))}$ for every ${g\in SL(2,\mathbb{R})}$ and ${x\in X}$.
• for each ${x\in X}$, there exists a constant ${c(x)>0}$ such that the cardinality ${N_V(x,R)}$ of the intersection ${V(x)\cap B(0, R)}$ of ${V(x)}$ with the ball ${B(0,R)}$ of center ${(0,0)}$ and radius ${R>0}$ is ${\leq c(x) R^2}$; moreover, the constant ${c(x)}$ can be chosen uniformly on compact subsets of ${X}$.
• there are ${R>0}$ and ${\varepsilon>0}$ such that the function ${N_V(x,R)}$ belongs to ${L^{1+\varepsilon}(X,\mu)}$.

The (non-trivial) fact that these conditions — especially the second and third items above — hold for the particular case of ${X=}$ a moduli space of translation surfaces and ${V=}$ 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 ${f\in C^{\infty}_0(\mathbb{R}^2)}$ of compact support on ${\mathbb{R}^2}$, let us define its Siegel-Veech transform ${\widehat{f}:X\rightarrow\mathbb{R}}$ as

$\displaystyle \widehat{f}(x)=\sum\limits_{v\in V(x)}f(v)$

In this language, the Siegel-Veech formula is

$\displaystyle \int_X \widehat{f}(x) \, d\mu(x) = c(\mu) \int_{\mathbb{R}^2} f(v) \, d\textrm{Leb}_{\mathbb{R}^2}(v)$

where ${c(\mu)=c_V(\mu)\geq 0}$ is the so-called Siegel-Veech constant of ${\mu}$ (with respect to ${V}$). At first sight, the Siegel-Veech formula looks tricky to prove (as ${X}$ and ${\mu}$ are “arbitrary”), but, as it turns out, this formula becomes easy to derive if we notice that

$\displaystyle f\in C^{\infty}_0(\mathbb{R}^2)\mapsto \int_X \widehat{f}(x) \, d\mu(x)$

is a non-negative linear functional on ${C^{\infty}_0(\mathbb{R}^2)}$, that is, the integration of Siegel-Veech transforms induces a measure on ${\mathbb{R}^2}$: indeed, this linear functional is well-defined because ${\widehat{f}}$ is finite, bounded on compact sets and ${\widehat{f}\in L^{1+\varepsilon}(X,\mu)\subset L^1(X,\mu)}$ by our assumptions on ${V}$. Furthermore, since ${V}$ varies linearly with ${SL(2,\mathbb{R})}$, it is not hard to see that this measure on ${\mathbb{R}^2}$ is ${SL(2,\mathbb{R})}$-invariant. Since the linear combinations of the Dirac measure at the origin ${(0,0)\in\mathbb{R}^2}$ and the Lebesgue measure ${\textrm{Leb}_{\mathbb{R}^2}}$ are the sole ${SL(2,\mathbb{R})}$-invariant measures on ${\mathbb{R}^2}$, it follows that this measure has the form

$\displaystyle \int_X \widehat{f}(x) \, d\mu(x) = a f(0,0)+ b\int f \, d\textrm{Leb}_{\mathbb{R}^2}$

Finally, since ${V(x)\subset\mathbb{R}^2-\{(0,0)\}}$, it is possible to check that ${a=0}$, so that the Siegel-Veech formula holds (with ${b=c_V(\mu)}$).

Once we know the Siegel-Veech formula, we can deduce that ${\mu(\mathcal{C}_1(\rho))=O(\rho^2)}$ by applying this formula with ${f_{\rho}}$ (a smooth “version” of) the characteristic function of the ball ${B(0,\rho)\subset\mathbb{R}^2}$:

$\displaystyle m(\mathcal{C}_1(\rho))\leq \int \widehat{f_{\rho}} \, d\mu = c(\mu) \int f_{\rho} \, d\textrm{Leb}_{\mathbb{R}^2}=O(\rho^2).$

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 ${SL(2,\mathbb{R})}$-action) saying that the Haar measures of the intersection of ${\mathcal{C}_2(\rho)}$ with certain pieces of ${SL(2,\mathbb{R})}$-orbits are ${o(\rho^2)}$. Then, we will use a sort of conditional measure argument to put together these “orbit by orbit estimates” to get the global estimate for ${\mu}$ in Theorem 1.

A little bit more precisely, our strategy is the following. Given a translation surface ${M}$, let ${\textrm{sys}(M)}$ denote the systole of ${M}$, that is, the length of the shortest saddle-connection(s). Given ${\rho>0}$, let ${X(\rho)=\{M\in\mathcal{C}:\textrm{sys}(M)=\rho\}}$. Inside the ${\rho}$-level ${X(\rho)}$ of the systole function ${\textrm{sys}}$, we consider the sets

$\displaystyle X_0^*(\rho):=\{M\in X(\rho): \textrm{ non-vertical saddle-connections have length }>\rho\}$

and

$\displaystyle X^*(\rho):=\bigcup\limits_{-\pi/2<\theta\leq\pi/2}R_{\theta}(X_0^*(\rho))$

where ${R_{\theta}\in SO(2,\mathbb{R})}$ denotes the rotation by ${\theta}$.

From the set ${X_0^*(\rho)}$, we can “access deeper levels” of the systole function via the set

$\displaystyle Y^*(\rho)=\bigcup\limits_{|\theta|<\pi/4}\bigcup\limits_{0\leq t<\log\cot|\theta|}g_t R_\theta(X_0^*(\rho))$

Indeed, the choice of ${\theta}$ and ${t}$ is guided by the fact that the vector ${g_t R_{\theta} e_2}$ is shorter than the (unit) vector ${e_2=(0,1)\in\mathbb{R}^2}$ for ${0\leq t<\log\cot|\theta|}$, ${|\theta|<\pi/4}$, so that the systole of ${g_t R_{\theta} M_0}$ is smaller than the systole of ${M_0\in X_0^*(\rho)}$.

Furthermore, ${Y^*(\rho)}$ is an interesting way to access ${\{M\in\mathcal{C}:\textrm{sys}(M)\leq\rho\}}$ because it is not hard to check that the sets ${g_t R_{\theta}(X_0^*(\rho))}$ for ${|\theta|<\pi/4}$ and ${0\leq t<\log\cot|\theta|}$ form a nice (measurable in the sense of Rokhlin) partition of ${Y^*(\rho)}$. In particular, by the ${SL(2,\mathbb{R})}$-invariance of ${m}$, we will be able to compute the ${m}$-measure of subsets of ${Y^*(\rho)}$ in terms of the Lebesgue measure ${dt}$ on ${\mathbb{R}}$, the Lebesgue measure ${\cos 2\theta d\theta}$ on the circle and a certain “density measure” ${m_0^{\rho}}$ on ${X_0^*(\rho)}$.

Using this disintegration, we can “transfer” mass from ${X_0^*(\rho)}$ to deep levels ${\{M\in\mathcal{C}:\textrm{sys}(M)\leq\rho\exp(-T)\}}$, ${T>0}$, as follows. Firstly, we will show that, for ${|\sin2\theta|<\exp(-2T)}$, there is an open interval ${J(T,\theta)}$ of ${t's}$ whose length is explicitly computable such that ${\textrm{sys}(g_t R_{\theta}(M_0))\leq\rho\exp(-T)}$ for all ${M_0\in X_0^*(\rho)}$. Geometrically, the set ${Y(\rho, T)}$ of ${g_tR_{\theta}M_0}$ for ${M_0\in X_0^*(\rho)}$, ${|\sin2\theta|<\exp(-2T)}$, ${t\in J(T,\theta)}$ correspond to the pieces of segments (“hyperbolas”) below the threshold ${\rho\exp(-T)}$.

Since the codimension 2 subsets ${R_{\theta}(X_0^*(\rho))}$ 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 ${m}$-measure of ${\{M\in\mathcal{C}:\textrm{sys}(g_t R_{\theta}(M))\leq\rho\exp(-T)\}}$ is at least

$\displaystyle m_0^{\rho}(X_0^*(\rho))\int_{|\sin2\theta|<\exp(-2T)} |J(T,\theta)| \cos2\theta d\theta=\frac{\pi}{2}(\exp(-T))^2 m_0^{\rho}(X_0^*(\rho))$

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

• as ${s\rightarrow 0}$, the ${m}$-measure of ${\{M\in\mathcal{C}:\textrm{sys}(g_t R_{\theta}(M))\leq s\}}$ is ${\frac{1}{2}(c(m)+o(1))s^2}$, and
• there exists a sequence ${(\rho_n)_{n\in\mathbb{N}}}$ with ${\rho_n\rightarrow 0}$ as ${n\rightarrow\infty}$ such that the densities ${\pi m_0^{\rho_n}(X_0^*(\rho_n))}$ are ${(c(m)-o(1))\rho_n^2}$.

Intuitively, this says that the densities of ${X_0^*(\rho_n)}$ 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 ${X_0^*(\rho)}$ is given by ${F'(\rho)/\rho}$ where ${F(\rho)=m(\{M\in\mathcal{C}: \textrm{sys}(M)\leq \rho\})}$. In particular, by L’Hôpital rule, we expect that

$\displaystyle \limsup\limits_{\rho\rightarrow0}F'(\rho)/\rho=c(m)$

if ${\lim\limits_{\rho\rightarrow0} F(\rho)/\rho^2=(1/2)c(m)}$.

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

$\displaystyle \begin{array}{rcl} \frac{1}{2}(c(m)+o(1))\rho_n^2\exp(-2T)&\geq& m(\{M\in\mathcal{C}:\textrm{sys}(M)\leq\rho_n\exp(-T)\}) \\ &\geq& m(Y(\rho_n, T))=\frac{\pi}{2}(\exp(-T))^2 m_0^{\rho_n}(X_0^*(\rho_n)) \\ &\geq& \frac{1}{2}(c(m)-o(1))(\rho_n\exp(-T))^2 \end{array}$

From this, we get that the set ${Y(\rho_n, T)}$ of translation surfaces with systole ${\leq \rho_n\exp(-T)}$ “accessed” from ${X_0^*(\rho_n)}$ occupies most of ${\{M\in\mathcal{C}:\textrm{sys}(M)\leq \rho_n\exp(-T)\}}$ in the sense that its complement has ${m}$-measure ${o(1)(\rho_n\exp(-T))^2}$ for all ${T>0}$.

Now, once we know that most translation surfaces with systole ${\leq \rho_n\exp(-T)}$ “come” from ${X_0^*(\rho)}$, we will complete the proof of Theorem 1 by showing that the translation surfaces ${M_0\in X_0^*(\rho_n)}$ leading to translation surfaces ${M=g_t R_{\theta} M_0\in Y(\rho_n, T)\cap \mathcal{C}_2(\rho_n\exp(-T))}$ are (essentially) those ${M_0}$ with two non-parallel saddle-connections of lengths comparable to ${\rho_n}$ making a very small angle ${\theta_0}$. Here, “very small angle” means that ${\theta_0}$ becomes close to zero for ${T}$ is sufficiently large (depending on ${\rho_n}$). Then, since the ${m_0^{\rho_n}}$-density of the set of those ${M_0}$ is small, say ${o(1)\rho_n^2}$, for ${\theta_0}$ small, i.e., ${T}$ large, we can use again that ${m}$ disintegrates as ${dt\times \cos2\theta d\theta\times m_0^{\rho_n}}$ to conclude that the ${m}$-measure of ${Y(\rho_n,T)\cap\mathcal{C}_2(\rho_n\exp(-T))}$ is ${o(1)(\rho_n\exp(-T))^2}$ for ${T}$ 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 ${SL(2,\mathbb{R})}$ allowing to compute the length ${|J(T,\theta)|}$ of the interval ${J(T,\theta)}$ 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” ${m_0^{\rho}}$ on ${X_0^*(\rho)}$. Finally, the last post will serve to formalize the “transport of mass” scheme described in the last 6 paragraphs above.

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