Posted by: matheuscmss | October 6, 2012

On sums of Lyapunov exponents of Kontsevich-Zorich cocycle (or an informal preparation for Pascal Hubert’s “Bourbaki seminar” talk on October 20, 2012)

In approximately 2 weeks, Pascal Hubert will deliver (on October 20 at Amphithéâtre Hermite of Institut Henri Poincaré (IHP) at 16h00) his “séminaire Bourbaki” lecture on this article of A. Eskin, M. Kontsevich and A. Zorich on the sums of Lyapunov exponents of the Kontsevich-Zorich cocycle with respect to {\textrm{SL}(2,{\mathbb R})}-invariant measures.

The goal of today’s post is to accomplish (below the fold) a promise made here of commenting on a few aspects of Eskin-Kontsevich-Zorich paper as an informal preparation to Pascal’s talk.

Remark 1 In this post we will use some parts of the following preliminary version of a survey by G. Forni and myself (currently with 150 pages) corresponding to our “Bedlewo lectures” at Banach center last year (to be submitted to the proceedings of the 2009 and 2011 editions of the School and Workshop “Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory” after Giovanni and I finish correcting it).Actually, this survey is the main reason why I temporarily stopped the series “Lyapunov spectrum of the Kontsevich-Zorich cocycle on the Hodge bundle of square-tiled cyclic covers” at its fourth installment: indeed, similarly to the “Homoclinic/heteroclinic bifurcations” series, I decided that it would be better to continue the series only when the corresponding survey text was ready because this would allow me to eventually post “expanded versions” of the sections of the survey article.

In particular, since the survey with Giovanni is almost finished, I hope to come back the “Lyapunov spectrum of the KZ cocycle on the Hodge bundle of square-tiled cyclic covers” after today’s post.

1. Introduction

The so-called Kontsevich-Zorich cocycle over the Teichmüller flow on the moduli space of unit area Abelian (and/or quadratic) differentials on Riemann surfaces is an object that we discussed several times in this blog (see, e.g., this series of posts here). In particular, we will not (re-)explain today to the reader the notions of Abelian differentials on Riemann surfaces, translation surfaces, moduli spaces of Abelian differentials and their strata, {\textrm{SL}(2,{\mathbb R})}-action and Teichmüller flow, Hodge bundle and Kontsevich-Zorich cocycle.

On the other hand, let us try to “convince” the reader that the Kontsevich-Zorich cocycle (KZ cocycle) is a natural object by briefly mentioning some of its applications:

Once we believe that the Kontsevich-Zorich cocycle is a fundamental object, let’s state the main formula in Eskin-Kontsevich-Zorich paper.

We start by considering a connected component {\mathcal{C}} of a stratum {\mathcal{H}^{(1)}(k_1,\dots, k_s)} of the moduli space {\mathcal{H}_g^{(1)}} of unit area Abelian differentials {\omega} on a Riemann surface {M}. From the intimate relationship between Abelian differentials and translation surfaces, we know that the natural (non-holomorphic!) action of {\textrm{SL}(2,{\mathbb R})} on {{\mathbb R}^2={\mathbb C}} “induces” a natural non-trivial {\textrm{SL}(2,{\mathbb R})}-action on {\mathcal{C}}. In this language, we recall that the Teichmüller flow is simply the action of the diagonal subgroup {g_t=\textrm{diag}(e^t,e^{-t})} of {\textrm{SL}(2,{\mathbb R})}.

The Hodge bundle {H^1_g} over {\mathcal{C}} is the vector bundle whose fiber over {(M,\omega)\in\mathcal{C}} is the first real cohomology group {H^1(M,\mathbb{R})}.

Remark 2 In fact, depending on the purposes, one may wish to consider the first real homology group {H_1(M,{\mathbb R})}, the first complex homology group {H_1(M,{\mathbb C})} or the first complex cohomology group {H^1(M,{\mathbb C})} as fibers of the Hodge bundle. In general, there is no big difference between these “Hodge bundles” because {H_1(M,{\mathbb R})} and {H^1(M,{\mathbb R})} are in duality and {H^1(M,{\mathbb C})=H^1(M,{\mathbb R})\oplus i H^1(M,{\mathbb R})}.

In this setting, the Kontsevich-Zorich cocycle {G_t^{KZ}} is the dynamical cocycle on the Hodge bundle {H^1_g} over the Teichmüller flow {g_t} obtained by transporting cohomology classes along Teichmüller flow orbits with the aid of the Gauss-Manin connection (for a more concrete description of KZ cocycle, see, e.g., this post here).

Remark 3 In order to be completely honest, let us point out that the Hodge bundle is not a vector bundle in the usual sense in Dynamical Systems (i.e., a vector bundle over a manifold) because the “orbifoldic nature” of moduli spaces. In particular, the KZ cocycle is not a dynamical cocycle (in general), but rather an orbifold cocycle in the sense that the linear transformations on the fibers of the Hodge bundle are defined up to a finite group (coming from the automorphisms of some Riemann surfaces)!

Today, we’ll be interested in the Lyapunov exponents of KZ cocycle. It is not hard to check from the definitions that KZ cocycle is symplectic with respect to the natural intersection form on the {2g}-dimensional vector space {H^1(M,{\mathbb R})} (where {g} is the genus of {M}). By Oseledets theorem, it follows that the Lyapunov spectrum of KZ cocycle (i.e., the collection of its {2g} Lyapunov exponents) with respect to any ergodic {g_t}-invariant probabilty {\mu} such that {G_{\pm1}^{KZ}} is {\log}-integrable (i.e., {\int_{\mathcal{C}}\|\log G_{\pm1}^{KZ}\|d\mu<\infty} for some measurable family of norms {\|.\|} on the fibers of {H^1_g}) has the form

\displaystyle \lambda_1^{\mu}\geq\lambda_2^{\mu}\geq\dots\geq\lambda_g^{\mu}\geq-\lambda_g^{\mu}\geq\dots\geq-\lambda_2^{\mu}\geq\lambda_1^{\mu}

Remark 4 By Remark 3, the “usual” statement of Oseledets theorem formally doesn’t apply to KZ cocycle in general because it is not a standard dynamical cocycle. However, since KZ cocycle differs from an usual dynamical cocycle by a finite group, it is not hard to see that the Lyapunov exponents still make sense and they are well-defined for the KZ cocycle.

Remark 5 As we will see later in this post (see Remark 9), there exists a continuous (family of) norm(s) on the fibers of {H_g^1} called Hodge norm(s) {\|.\|_{Hodge}} such that {\log\|G_{\pm1}^{KZ}\|_{Hodge}\leq 1}. In particular, we have that {G_t^{KZ}} is {\log}-integrable with respect to any {g_t}-invariant probability {\mu}. In other words, the {\log}-integrability of hypothesis in Oseledets theorem is “automatically” satisfied in the case of KZ cocycle.

In their article, A. Eskin, M. Kontsevich and A. Zorich showed the following fundamental result for the sum {\lambda_1^{\mu}+\dots+\lambda_g^{\mu}} of the non-negative Lyapunov exponents of an ergodic {\textrm{SL}(2,{\mathbb R})}-invariant probability {\mu}:

Theorem 1 (Eskin-Kontsevich-Zorich) Let {\mu} be an ergodic {\textrm{SL}(2,{\mathbb R})}-invariant probability on a connected component {\mathcal{C}} of a stratum {\mathcal{H}^{(1)}(k_1,\dots, k_s)} of the moduli space of unit area Abelian differentials of genus {g}. Then,

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

where {c(\mu)} is a geometric quantity called Siegel-Veech constant of {\mu}.

Remark 6 Again, in order to be completely honest, we must say that, in their article, A. Eskin, M. Kontsevich and A. Zorich stated their result only for the ergodic {\textrm{SL}(2,\mathbb{R})}-invariant probabilities that are “affine” and “regular”. As it turns out, A. Eskin and M. Mirzakhani recently proved (the “Ratner-like theorem”) that all ergodic {\textrm{SL}(2,\mathbb{R})}-invariant probabilities are affine. Furthermore, all known ergodic {\textrm{SL}(2,\mathbb{R})}-invariant probabilities are regular. In any event, in this post we’ll never use the “affine” and “regular” assumptions, so that we’ll forget about this remark right away.

Remark 7 There is an analog formula (also due to Eskin-Kontsevich-Zorich) for the case of quadratic differentials. However, for the sake of simplicity, we’ll deal exclusively with Abelian differentials here.

Of course, this statement of Eskin-Kontsevich-Zorich theorem (EKZ theorem for short) is incomplete because we don’t know yet the definition of the Siegel-Veech constant {c(\mu)}. In principle, we could give it right now, but we’ll do so only in the last section of this post. Here, our “motivation” for doing so is the following one. Very roughly speaking, the first step towards the proof of EKZ theorem is the proof of the so-called Kontsevich-Forni formula, a result presenting {\lambda_1^{\mu}+\dots+\lambda_g^{\mu}} as an integral

\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\int_{\mathcal{C}}\Lambda d\mu \ \ \ \ \ (1)

of a certain function {\Lambda} on {\mathcal{C}} with respect to {\mu}. As it turns out, for this first step (i.e., Kontsevich-Forni formula), one doesn’tneed the Siegel-Veech constants and, in fact, these quantities are relevant only when one wants to convert the integral {\int_{\mathcal{C}}\Lambda d\mu} into a more tractable object (i.e., when one wants to actually compute the integral). Thus, for this reason, we will divide this post as follows: in the next section, we will state and sketch the proof of Kontsevich-Forni formula, and in the last section we will define the Siegel-Veech constants and give some applications of EKZ theorem.

Remark 8 In some sense, the Kontsevich-Forni formula “explains” why the right-hand side of EKZ formula has two contributions ({\frac{1}{12}\sum\limits_{i=1}^s\frac{k_i(k_i+2)}{k_i+1}} and {c(\mu)}). Indeed, if the moduli space (or rather {\mathcal{C}}) were compact, an application of Stokes theorem and analytic Grothendieck-Riemann-Roch formula would show that the integral equals {\frac{1}{12}\sum\limits_{i=1}^s\frac{k_i(k_i+2)}{k_i+1}}. However, the non-compactness of {\mathcal{C}} makes things more complicated because, besides {\frac{1}{12}\sum\limits_{i=1}^s\frac{k_i(k_i+2)}{k_i+1}}, we get a “boundary” contribution that, after a lot of work, one can show that is equal to the “geometrically significant quantity” {c(\mu)}.

2. Kontsevich-Forni formula

By definition, the task of studying Lyapunov exponents consists precisely in understanding the growth of norm of vectors. Of course, the particular choice of norm doesn’t affect the values of Lyapunov exponents (essentially because two norms on a finite-dimensional vector space are equivalent), but for the sake of our discussion it will be convenient to work with the so-called Hodge norm.

2.1. Hodge norm on the Hodge bundle over {\mathcal{H}_g^{(1)}}

Let {M} be a Riemann surface. The Hodge (intersection) form {(.,.)} on {H^1(M,\mathbb{C})} is given

\displaystyle (\alpha,\beta):=\frac{i}{2}\int_M \alpha\wedge\overline{\beta}

for each {\alpha, \beta\in H^1(M,\mathbb{C})}.

The Hodge form is positive-definite on the space {H^{1,0}(M)} of holomorphic {1}-forms on {M}, and negative-definite on the space {H^{0,1}(M)} of anti-holomorphic {1}-forms on {M}. For instance, given a holomorphic {1}-form {\alpha\neq0}, we can locally write {\alpha(z)=f(z)dz}, so that

\displaystyle \alpha(z)\wedge\overline{\alpha(z)}=|f(z)|^2 dz\wedge\overline{dz} = -2i|f(z)|^2 dx\wedge dy.

Since {dx\wedge dy} is an area form on {M} and {|f(z)|^2\geq 0}, we get that {(\alpha,\alpha)>0}.

In particular, since {H^1(M,\mathbb{C})=H^{1,0}(M)\oplus H^{0,1}(M)}, and {H^{1,0}(M)} and {H^{0,1}(M)} are {g}-dimensional complex vector spaces, one has that the Hodge form is an Hermitian form of signature {(g,g)} on {H^1(M,\mathbb{C})}.

The Hodge form induces an Hermitian form (also called Hodge form and denoted by {(.,.)=(.,.)_{Hodge}}) on the complex Hodge bundle {H_g^1(\mathbb{C})}.

The so-called Hodge representation theorem says that any real cohomology class {c\in H^1(M,\mathbb{R})} is the real part of an unique holomorphic {1}-form {h(c)\in H^{1,0}(M)}, i.e., {c=\Re h(c)}. In particular, one can use the Hodge form {(.,.)} to induce an inner product on {H^1(M,\mathbb{R})} via:

\displaystyle (c_1,c_2):=(\Re h(c_1), \Re h(c_2))

for each {c_1, c_2\in H^1(M,\mathbb{R})}.

Again, this induces an inner product {(.,.)=(.,.)_{Hodge}} and a norm {\|.\|=\|.\|_{Hodge}} on the real Hodge bundle {H_g^1(\mathbb{R})}. In the literature, {(.,.)} is the Hodge inner product and {\|.\|} is the Hodge norm on the real Hodge bundle.

Observe that, in general, the subspaces {H^{1,0}} and {H^{0,1}} are not equivariant with respect to the (natural complex version of the) KZ cocycle (on {H_g^1(\mathbb{C})}), and this is one of the reasons why the Hodge norm {\|.\|} is not preserved by the KZ cocycle in general. In the next subsection, we will study first variation formulas for the Hodge norm along the KZ cocycle and its applications to the Teichmüller flow.

2.2. {1}st variation of Hodge norm

Let {c\in H^1(M,\mathbb{R})} a vector in the fiber of the real Hodge bundle over {\omega\in\mathcal{C}}. Denote by {\alpha_0} the holomorphic {1}-forms with {c=\Re\alpha_0}. By applying the Teichmüller flow {g_t} to {\omega}, we endow {M} with a new Riemann surface structure such that {\omega_t=g_t(\omega)} is an Abelian differential. In particular, {c=\Re\alpha_t} where {\alpha_t} is a holomorphic {1}-form with respect to the new Riemann surface structure associated to {\omega_t}.

Of course, by definition, KZ cocycle acts by parallel transport (with respect to Gauss-Manin connection) on the Hodge bundle, so that the cohomology classes {c} are not “changing”. However, since the representatives {\alpha_t} we use to “measure” the “size” (Hodge norm) of {c} are changing, it is an interesting (and natural) problem to know how fast the Hodge norm changes along KZ cocycle, or, equivalently, to compute the first variation of the Hodge norm along KZ cocycle:

\displaystyle \frac{d}{dt}\|\pi_2(G_t^{KZ}(\omega,c)\|_{\omega_t}^2|_{t=0}:=\frac{d}{dt}\|c\|_{\omega_t}:=\frac{d}{dt}(\alpha_t,\alpha_t)|_{t=0}

where {\pi_2:H_g^{(1)}\rightarrow H^1(M,\mathbb{R})} is the projection in the fibers of the Hodge bundle and {\|.\|_{\omega_t}} is the Hodge norm with respect to the Riemann surface structure induced by {\omega_t}.

In this subsection we will calculate this quantity by following (Section 2 of) the original article of G. Forni (see also the recent survey by G. Forni, A. Zorich and myself where a differential-geometric interpretations of Giovanni’s arguments are given).

By working locally outside the zeroes of {\omega_t}, we can choose local holomorphic coordinates {z_t} with {\omega_t=dz_t}. Now, we note that, by definition of the Teichmüller flow, {dz_t=\omega_t := e^t dx+ie^{-t}dy}, where {dz=dx+idy}, so that

\displaystyle \dot{\omega_t}:=\frac{d}{dt}\omega_t = e^t [\Re\omega] - i e^{-t} [\Im\omega]=\overline{\omega_t}=d\overline{z_t}.

Next, we write {0=c-c=[\Re\alpha_t]-[\Re\alpha_0]}, so that we find smooth family {u_t} with {du_t=\Re\alpha_t-\Re\alpha_0}. By writing {\alpha_t=f_t\omega_t}, and by taking derivatives, we have locally

\displaystyle d\dot{u_t}=\dot{f_t}\omega_t+f_t\dot{\omega_t}+\overline{\dot{f_t}}\overline{\omega_t}+\overline{f_t}\overline{\dot{\omega_t}} = \dot{f_t}dz_t + f_t d\overline{z_t}+\overline{\dot{f_t}}d\overline{z_t}+\overline{f_t}dz_t

In particular, since {(\partial \dot{u_t}/\partial z_t)dz_t + (\partial \dot{u_t}/\partial \overline{z_t})d\overline{z_t} := d\dot{u_t}}, we find that {\partial \dot{u_t}/\partial z_t= \dot{f_t}+\overline{f_t}}.

Finally, we can (locally) compute

\displaystyle \begin{array}{rcl} \frac{d}{dt}(\alpha_t,\alpha_t) &=& \Re\frac{i}{2} \frac{d}{dt} \int \alpha_t\wedge\overline{\alpha_t} = \Re\frac{i}{2} \frac{d}{dt} \int f_t\overline{f_t}(\omega_t\wedge\overline{\omega_t}) = \Re\frac{i}{2} \frac{d}{dt} \int f_t\overline{g_t}(dz_t\wedge d\overline{z_t}) \\ &=& 2\Re\frac{i}{2} \int f_t\overline{\dot{f_t}}(dz_t\wedge d\overline{z_t}) = 2\Re\frac{i}{2}\int f_t\left(-\overline{\overline{f_t}} + \frac{\partial \overline{\dot{u_t}}}{\partial \overline{z_t}}\right)dz_t\wedge d\overline{z_t} \\ &=&-2\Re \frac{i}{2}\int f_t f_t (dz_t\wedge d\overline{z_t}) = -2\Re \frac{i}{2}\int \frac{\alpha_t\cdot\alpha_t}{\omega_t} \omega_t\wedge \overline{\omega_t} \end{array}

In resume, we proved the following {1}st variation formula (originally from Lemma 2.1′ of Forni’s 2002 paper):

Theorem 2 (G. Forni) Let {\omega} be an Abelian differential and {c\in H^1(M,\mathbb{R})}. Denote by {\alpha_0} the holomorphic (with respect to {(M,\omega)}) {1}-form with {c=[\Re\alpha_0]}. Then,

\displaystyle \frac{d}{dt}\|\pi_2(G_t^{KZ}(\omega,c)\|_{\omega_t}^2|_{t=0} = \frac{d}{dt}\|c\|_{\omega_t}^2|_{t=0} = -2\Re B_{\omega}(\alpha_0,\alpha_0)

where {B_{\omega}(\alpha,\beta):=\frac{i}{2} \int_M (\alpha/\omega)(\beta/\omega)\omega\wedge\overline{\omega}}.

In order to aleviate the notation, we put {B_{\omega}^{\mathbb{R}}(c,c):=B_{\omega}(\alpha_0,\alpha_0)} where {\alpha_0} is the unique {(M,\omega)}-holomorphic {1}-form with {c=[\Re\alpha_0]}. Observe that {B_{\omega}^{\mathbb{R}}} is a complex-valued bilinear form.

Corollary 3 One has

\displaystyle \frac{d}{dt}\log\|c\|_{\omega_t}|_{t=0} = -\frac{\Re B^{\mathbb{R}}_{\omega}(c,c)}{\|c\|_{\omega}^2}

In particular,

\displaystyle \frac{d}{dt}\log\|c\|_{\omega_t}|_{t=0}\leq 1

Proof: The first statement of this corollary follows from the main formula in Theorem 2, while the second statement follows from an application of Cauchy-Schwarz inequality:

\displaystyle \begin{array}{rcl} |B_{\omega}(\alpha,\beta)|&\leq& \int\left|(\alpha/\omega)(\beta/\omega)\right|\omega\wedge\overline{\omega}\leq \left(\int |\alpha/\omega|^2 \omega\wedge\overline{\omega}\right)^{1/2}\left(\int |\beta/\omega|^2 \omega\wedge\overline{\omega}\right)^{1/2} \\ &=& \|\alpha\|_{\omega}\|\beta\|_{\omega} \end{array}

\Box

Remark 9 This corollary implies that the KZ cocycle is {\log}-bounded with respect to the Hodge norm, that is, {\log\|G_{\pm1}^{KZ}(\omega,c)\|_{g_{\pm1}(\omega)}\leq 1} for all {c\in H^1(M,\mathbb{R})} with {\|c\|_{\omega}=1}. Hence, given any finite mass measure {\mu} on {\mathcal{H}_g^{(1)}}, we have that

\displaystyle \int\log^+\|G_{\pm1}^{KZ}(\omega)\|d\mu<\infty

Corollary 4 Let {\mu} be any {g_t}-invariant ergodic probability on {\mathcal{C}}. Then, {\lambda_2^{\mu}<1=\lambda_1^{\mu}}.

Proof: By Corollary 3, we have that {\lambda_1^{\mu}\leq 1}. Moreover, since the Teichmüller flow {g_t(\omega):=e^t\Re\omega+ i e^{-t}\Im\omega=\omega_t}, we have that the {G_t^{KZ}}-invariant {2}-plane {H^1_{st}(M,\mathbb{R}):=\mathbb{R}\cdot[\Re\omega]\oplus \mathbb{R}\cdot[\Im\omega]} contributes with Lyapunov exponents {\pm1}. In particular, {\lambda_1^{\mu}=1}.

Now, we note that {H^1_{(0)}(M,\mathbb{R}):=\{c\in H^1(M,\mathbb{R}):c\wedge\omega=0\}} is {G_t^{KZ}}-invariant because the KZ cocycle is symplectic with respect to the intersection form on {H^1(M,\mathbb{R})} and {H^1_{(0)}(M,\mathbb{R})} is the symplectic orthogonal of the (symplectic) {2}-plane {H^1_{st}(M,\mathbb{R})}. Therefore, {\lambda_2^{\mu}} is the largest Lyapunov exponent of the restriction of KZ cocycle to {H^1_{(0)}(M,\mathbb{R})}.

In order to estimate {\lambda_2^{\mu}}, we observe that, for any {c\in H^1_{(0)}(M,\mathbb{R})-\{0\}},

\displaystyle \frac{d}{dt}\log\|c\|_{\omega_t}|_{t=0} = -\frac{\Re B^{\mathbb{R}}_{\omega}(c,c)}{\|c\|_{\omega}^2}\leq \Lambda^+(\omega):= \max\left\{\frac{|B^{\mathbb{R}}_{\omega}(h,h)|}{\|h\|_{\omega}^2}: h\in H^1_{(0)}(M,\mathbb{R})-\{0\}\right\}

by Corollary 3. Hence, by integration,

\displaystyle \frac{1}{T}(\log\|c\|_{g_T(\omega)}-\log\|c\|_{\omega})\leq\frac{1}{T}\int_0^T\Lambda^+(g_t(\omega))dt

By Oseledets theorem and Birkhoff’s theorem, for {\mu}-almost every {\omega\in\mathcal{C}}, we obtain that

\displaystyle \lambda_2^{\mu}=\lim\limits_{T\rightarrow\infty}\frac{1}{T}\log\|c\|_{g_T(\omega)}\leq\lim\limits_{T\rightarrow\infty}\frac{1}{T}\int_0^T\Lambda^+(g_t(\omega))dt = \int_{\mathcal{C}} \Lambda^+(\omega) d\mu(\omega)

This reduces the task of proving that {\lambda_2^{\mu}<1} to show that {\Lambda^+(\omega)<1} for every {\omega\in\mathcal{C}}. Here, we proceed by contradiction. Assume that {\Lambda^+(\omega)=1} for some Abelian differential {\omega}. By definition, this means that

\displaystyle |B_{\omega}^{\mathbb{R}}(h,h)|=\|h\|_{\omega}^2

for some {h\in H^1_{(0)}(M,\mathbb{R})-\{0\}}. In other words, by looking at the proof of Corollary 3, we have a case of equality in an estimate derived from Cauchy-Schwarz inequality. It follows that, by denoting {\alpha(h)\neq 0} the {(M,\omega)}-holomorphic {1}-form with {h=[\Re\alpha(h)]}, the functions {u(h):=\alpha(h)/\omega} and {\overline{u(h)}=\overline{\alpha(h)}/\overline{\omega}} differ by a multiplicative constant {a\in\mathbb{C}}, i.e.,

\displaystyle \overline{u(h)}=a\cdot u(h)

Since {u(h)} is a meromorphic function and, a fortiori, {\overline{u(h)}} is an anti-meromorphic function, this is only possible when {u(h)} is a constantfunction, that is, {\alpha(h)\in\mathbb{C}\cdot\omega-\{0\}}. In particular, {h\wedge \omega\neq 0}, a contradiction with the fact that {h\in H^1_{(0)}(M,\mathbb{R})}. \Box

By a careful inspection of the proof of the previous corollary, we saw that {\lambda_2^{\mu}\leq \int_{\mathcal{C}} \Lambda^+(\omega) d\mu(\omega)}, i.e., the second Lyapunov exponent is naturally bounded by the integral of a certain (continuous) function {\Lambda^+} on {\mathcal{C}}. In some sense, this is a “preview” of the Kontsevich-Zorich formula (compare with (1)) that we will state in the next subsection.

Remark 10 At this stage, one could work more to derive further applications of the Hodge norm to Teichmüller dynamics: for instance, using the Hodge norm it is possible to show some uniform hyperbolicity and quantitative recurrence estimates for the Teichmüller flow {g_t} with respect to any compact set {K\subset \mathcal{C}}, and this information was used by J. Athreya and G. Forni to study deviations of ergodic averages for billiards on rational polygons. However, we will refrain ourselves from doing so because it would lead us too far away from Kontsevich-Forni formula.

2.3. {2}nd variation of Hodge norm and Kontsevich-Zorich formula

Geometrically, {B_{\omega}} is essentially the second fundamental form (or Kodaira-Spencer map) of the holomorphic subbundle {H^{1,0}} of the complex Hodge bundle {H^1_{\mathbb{C}}} equipped with the Gauss-Manin connection. Roughly speaking, recall that the second fundamental form {II_{\omega}:H^{1,0}\rightarrow H^{0,1}} is {II_{\omega}(c)=\frac{d}{dt}c_t^{0,1}} where {c_t^{0,1}} is the {H^{0,1}}-component of {G_t^{KZ}(c)}. See the figure below.

In this language, it is possible to show that {B_{\omega}(\alpha,\beta)=-(\overline{II_{\omega}(\alpha)},\beta)} where {(.,.)} is the Hodge form. See, e.g., this survey for more discussion on this differential-geometrical interpretation of {B}. (A word of caution: the second fundamental form {A_{\omega}(c)} considered herediffers from {II_{\omega(c)}} by a sign, i.e., {II_{\omega}(c)=-A_{\omega}(c)}!)

Next, by taking {\{\omega_1,\dots,\omega_g\}} a Hodge-orthonormal basis of {H^{1,0}}, we have a matrix {B=(B_{jk})_{1\leq j,k\leq g}}, {B_{jk}=\frac{i}{2}\int\frac{\omega_j\omega_k}{\omega}\overline{\omega}}, associated to {B_{\omega}}. Define {H=H_{\omega}:=B\cdot B^*}. The eigenvalues {\Lambda_1(\omega)\geq\dots\geq \Lambda_g(\omega) (\geq 0)} of {H} have the form {|\lambda|^2} where {\lambda} is an eigenvalue of {B}, i.e., {H} induces a positive semi-definite form on {H^{1,0}}. As it turns out, {H} is essentially the curvature form of the holomorphic subbundle {H^{1,0}} of the complex Hodge bundle {H^1_{\mathbb{C}}} equipped with the Gauss-Manin connection (see this survey for more details), i.e., the matrix {H} also a differential-geometrical interpretation (similarly to {B}). In particular, this geometrical interpretation hints that {H} should naturally enter into second variation formulas for the Hodge norm (of course, this should be compared with the fact that {B} naturally enters into first variation formulas for the Hodge norm) and, a fortiori, the eigenvalues of {H} should provide nice consequences to the study of Lyapunov exponents. In fact, as it was proposed by M. Kontsevich and proved by G. Forni in 2002, one can relate the eigenvalue of {H} to Lyapunov exponents of KZ cocycle via the following formula:

Theorem 5 (M. Kontsevich, G. Forni) Let {\mu} be a {SL(2,\mathbb{R})}-invariant {g_t}-ergodic probability on a connected component {\mathcal{C}} of some stratum of {\mathcal{H}_g^{(1)}}. Then, one has the following formula for the sum of non-negative Lyapunov exponents of KZ cocycle with respect to {\mu}:

\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\int_{\mathcal{C}}(\Lambda_1(\omega)+\dots+\Lambda_g(\omega))\,d\mu(\omega)

Remark 11 Since {B_{\omega}(\omega,\omega):=1}, one can use the argument (Cauchy-Schwarz inequality) of the proof of Corollary 3 to see that {\Lambda_1(\omega)\equiv 1} for all {\omega}. In particular, since {\lambda_1^{\mu}=1}, one can rewrite the formula above as

\displaystyle \lambda_2^{\mu}+\dots+\lambda_g^{\mu}=\int_{\mathcal{C}}(\Lambda_2(\omega)+\dots+\Lambda_g(\omega))\,d\mu(\omega)

Remark 12 Note that there is an important difference in the hypothesis of Theorem 2 and Theorem 5 is: in the former {\mu} is any {g_t}-invariant while in the latter {\mu} is {SL(2,\mathbb{R})}-invariant!

Let’s give now an outline of proof of Theorem 5. Given {1\leq k\leq g}, let

\displaystyle \Phi_k(\omega, I_k):=2\sum\limits_{i=1}^k H_{\omega}(c_i, c_i) - \sum\limits_{j,m=1}^k |B_{\omega}^{\mathbb{R}}(c_j,c_m)|^2

where {I_k} is a {k}-dimensional isotropic subspace of the real Hodge bundle {H^1_{\mathbb{R}}} and {\{c_1,\dots,c_k\}} is anyHodge-orthonormal basis of {I_k}.

Remark 13 Of course, it is implicit here that the expression

\displaystyle 2\sum\limits_{i=1}^k H_{\omega}(c_i, c_i) - \sum\limits_{j,m=1}^k |B_{\omega}^{\mathbb{R}}(c_j,c_m)|^2

doesn’t depend on the choice of Hodge-orthonormal basis {\{c_1,\dots,c_k\}} but only on the isotropic subspace {I_k\subset H^1(M,\mathbb{R})}. We leave this verification as an exercise to the reader.

In the sequel, we will use the following three lemmas (see Forni’s 2002 paper or this survey for proofs and more details).

Lemma 1 (Lemma 5.2′ of Forni’s 2002 paper). Let {\{c_1,\dots,c_k,c_{k+1},\dots,c_g\}} be any Hodge-orthonormal completion of {\{c_1,\dots,c_k\}} into basis of a Lagrangian subspace of {H^1(M,\mathbb{R})}. Then,

\displaystyle \Phi_k(\omega, I_k)=\sum_{i=1}^g\Lambda_i(\omega)-\sum\limits_{j,m=k+1}^g|B^{\mathbb{R}}_{\omega}(c_j,c_m)|^2

Remark 14 (M. Kontsevich’s fundamental remark) In the extremal case {k=g}, the right-hand side of the previous equality doesn’tdepend on the Lagrangian subspace {I_g}:

\displaystyle \Phi_g(\omega, I_g)=\sum_{i=1}^g\Lambda_i(\omega) = \textrm{tr}(H_{\omega})

This fundamental observation (that is hard to overestimate!) of Maxim Kontsevich lies at the heart of the main formula of Theorem 5

It is not hard to see that the notion of Hodge norm {\|.\|_{\omega}} on vectors {c\in H^1(M,\mathbb{R})} can be extended to any polyvector {c_1\wedge\dots\wedge c_k} coming from a (Hodge-orthonormal) basis {\{c_1,\dots,c_k\}} of an isotropic subspace {I_k}. By slightly abusing of the notation, we will denote by {\|c_1\wedge\dots\wedge c_k\|_{\omega}} the Hodge norm of such a polyvector.

Note that the Hodge norm {\|.\|_{\omega}} depends only on the complex structure, so that {\|.\|_{\omega}=\|.\|_{\omega'}} whenever {\omega'=\textrm{constant}\cdot\omega}. In particular, it makes sense to consider the Hodge norm {\|.\|_h} over the Teichmüller disk {h\in SO(2,\mathbb{R})\backslash SL(2,\mathbb{R})\cdot\omega}. For subsequent use, we denote by {\Delta_{hyp}} the hyperbolic (leafwise) Laplacian on {SO(2,\mathbb{R})\backslash SL(2,\mathbb{R})\cdot\omega} (here, we’re taking advantage of the fact that {SO(2,\mathbb{R})\backslash SL(2,\mathbb{R})} is isomorphic to Poincaré’s hyperbolic disk {D}).

Lemma 2 (Lemma 5.2 of Forni’s 2002 paper). One has {\Delta_{hyp}\log\|c_1\wedge\dots\wedge c_k\|_{\omega} = 2\Phi_k(\omega, I_k)}.

Finally, in order to connect the previous two lemmas with Oseledets theorem (and Lyapunov exponents), one needs the following fact about hyperbolic geometry:

Lemma 3 (Lemma 3.1 of Forni’s 2002 paper). Let {L:D\rightarrow\mathbb{R}} be a smooth function. Then,

\displaystyle \frac{1}{2\pi}\frac{\partial}{\partial t}\int_0^{2\pi} L(t,\theta)\,d\theta = \frac{1}{2}\tanh(t)\frac{1}{\textrm{area}(D_t)}\int_{D_t}\Lambda\, d\textrm{area}_P

where {\Lambda:=\Delta_{hyp} L}, {(t,\theta)} are polar coordinates on Poincaré’s disk, {D_t} is the disk of radius {t} centered at the origin {0\in D} and {\textrm{area}_P} is Poincaré’s area form on {D}.

Next, the idea to derive Theorem 5 from the previous three lemmas is the following. Denote by {R_\theta=\left(\begin{array}{cc}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{array}\right)}, and, given {\omega\in\mathcal{H}_g^{(1)}}, for {SO(2,\mathbb{R})\backslash SL(2,\mathbb{R})\cdot\omega\ni h=g_tR_{\theta}\omega=(t,\theta)}, let {L(h):=\|c_1\wedge\dots\wedge c_k\|_h}. In plain terms, {L} is measuring how the (Hodge norm) size of the polyvector {c_1\wedge\dots\wedge c_k} changes along the Teichmüller disk of {\omega}. In particular, as we’re going to see in a moment, it is not surprising that {L} has “something to do” with Lyapunov exponents.

By Lemma 3, one has

\displaystyle \frac{1}{2\pi}\frac{\partial}{\partial t}\int_0^{2\pi} L(t,\theta)\,d\theta = \frac{1}{2}\tanh(t)\frac{1}{\textrm{area}(D_t)}\int_{D_t}\Delta_{hyp}L(t,\theta)\, d\textrm{area}_P

Then, by integrating with respect to the {t}-variable in the interval {[0,T]} and by using Lemma 2 for the computation of {\Delta_{hyp}L}, one deduces

\displaystyle \frac{1}{2\pi}\frac{1}{T}\int_0^{2\pi} (L(T,\theta)-L(0,\theta))\,d\theta = \frac{1}{T}\int_0^T\frac{\tanh(t)}{\textrm{area}(D_t)}\int_{D_t}\Phi_k(t,\theta)\, d\textrm{area}_P\,dt

At this point, by taking an average with respect to {\mu} and using the {SL(2,\mathbb{R})}-invariance of {\mu} to get rid of the integration with respect to {\theta}, we deduce that

\displaystyle \frac{1}{T}\int_{\mathcal{C}}(L(g_T(\omega))-L(\omega))\,d\mu(\omega) = \frac{1}{T}\int_{\mathcal{C}}\int_0^T\frac{\tanh(t)}{\textrm{area}(D_t)}\int_{D_t}\Phi_k(g_tR_{\theta}\omega, I_k)\, d\textrm{area}_P\,dt\, d\mu(\omega)

Now, we observe that:

  • by Oseledets theorem, for a “generic” isotropic subspace {I_k} and {\mu}-almost every {\omega}, one has that {\frac{1}{T}L(g_T(\omega))} converges to {\lambda_1^{\mu}+\dots+\lambda_k^{\mu}} as {T\rightarrow \infty} (recall that, by definition, the function {t\mapsto L(g_t(\omega))} is measuring the growth (in Hodge norm) of the polyvector {c_1\wedge\dots\wedge c_k} along the Teichmüller orbit {g_t(\omega)}), and
  • by Remark 14, for {k=g}, {\Phi_g(\omega, I_g)=\Phi_g(\omega)=\Lambda_1(\omega)+\dots+\Lambda_g(\omega)} is independent on {I_g}.

So, for {k=g}, this discussion (combined with an application of Lebesgue dominated convergence theorem and the fact that {\tanh(t)/\textrm{area}(D_t)\rightarrow 1} as {t\rightarrow\infty}. See, e.g., this survey for more details) allows to show that

\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\int_{\mathcal{C}}(\Lambda_1(\omega)+\dots+\Lambda_g(\omega))\,d\mu(\omega)

This completes the sketch of proof of Theorem 5.

Remark 15 Essentially the same argument above allows to derive formulas for partial sumsof Lyapunov exponents. More precisely, given {\mu} a {SL(2,\mathbb{R})}-invariant {g_t}-ergodic probability with {\lambda_k^{\mu}>\lambda_{k+1}^{\mu}} (for some 1\leq k\leq g-1), one has

\displaystyle \lambda_1^{\mu}+\dots+\lambda_k^{\mu}=\int_{\mathcal{C}} \Phi_k(\omega,E^+_k(\omega))\,d\mu(\omega)

where {E^+_k(\omega)} is the Oseledets subspace associated to the {k} top Lyapunov exponents. In general, this formula is harder to use than Theorem 5 because the right-hand side of the former implicitly assumes some a priori control of {E^+_k(\omega)} while the right-hand side of the latter is independent of Lagrangian subspaces (as noticed by M. Kontsevich).

Remark 16 Actually, for the “integration by parts” argument relevant for the EKZ formula, the most “convenient” form of Kontsevich-Forni formula is

\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\int_{\mathcal{C}}\frac{1}{2}\Delta_{hyp}\log\|c_1\wedge\dots\wedge c_g\|_{\omega}\,d\mu(\omega)

where {c_1,\dots, c_g} is a Hodge-orthogonal basis of a Lagrangian subspace of {H^1(M,{\mathbb R})}.

Now, let us complete the discussion of today’s post by making some comments in the next (final) section on Siegel-Veech constants and some applications of EKZ formula.

3. Siegel-Veech constants and EKZ formula

3.1. Siegel-Veech constants

Given an Abelian differential {(M,\omega)}, we consider the associated translation surface. Given a closed regular geodesic {\gamma} in a translation surface {(M,\omega)}, we can form a maximal cylinder {C} by collecting all closed geodesics of {(M,\omega)} parallel to {\gamma} not meeting any zero of {\omega}. In particular, the boundary of {C} contains zeroes of {\omega}. Given a maximal cylinder {C}, we denote by {w(C)} its width (i.e., the length of its waist curve {\gamma}) and by {h(C)} its height (i.e., the distance across {C}). For example, in the figure below we illustrate two closed geodesics {\gamma_1} and {\gamma_2} (in the horizontal direction) and the two corresponding maximal cylinders {C_1} and {C_2} of a L-shaped square-tiled surface (see these posts here and here for more explanations). In this picture, we see that {C_1} has width {2}, {C_2} has width {1}, and both {C_1} and {C_2} have height {1}.

Definition 6 Let {(M,\omega)} be a translation surface. Given {L>0}, we define

\displaystyle N_{\textrm{area}}(\omega,L)=\sum\limits_{\substack{C \textrm{ maximal horizontal cylinder } \\ \textrm{ of width }w(C)<L}}\frac{\textrm{area}(C)}{\textrm{area}(\omega)}

Informally, {N_{\textrm{area}}(\omega, L)} counts the fraction of the area of the translation surface {(M,\omega)} occupied by maximal horizontal cylinders of width bounded by {L}.

Of course, the quantity {N_{\textrm{area}}(\omega, L)} depends a lot on the geometry of {(M,\omega)} and the real number {L>0}. However, W. Veech and Ya. Vorobets discovered that given any {SL(2,\mathbb{R})}-invariant {g_t}-ergodic probability {\mu}, the quantity

\displaystyle c(\mu)=\frac{\pi}{3L^2}\int N_{\textrm{area}}(\omega, L) d\mu(\omega)

doesn’t depend on {L>0}. In the literature, {c(\mu)} is called the Siegel-Veech constantof {\mu}.

Remark 17 Our choice of normalization of the quantity {\frac{1}{L^2}\int N_{\textrm{area}}(\omega, L) d\mu(\omega)} leading to the Siegel-Veech constant here is not the same of Eskin-Kontsevich-Zorich. Indeed, what they call Siegel-Veech constant is {3c(\mu)/\pi^2} in our notation. Of course, there is no conceptual different between these normalizations, but we prefer to take a different convention from Eskin-Kontsevich-Zorich because {c(\mu)} appears more “naturally” in the statement of EKZ formula.

Remark 18 It is not hard to see from the definition that Siegel-Veech constants {c(\mu)} are always positive, i.e., {c(\mu)>0} for any {SL(2,\mathbb{R})}-invariant {g_t}-ergodic {\mu}.

The Siegel-Veech constants of Masur-Veech measures {\mu_{MV}} were computed by A. Eskin, H. Masur and A. Zorich and they are intimately related to volumes {\mathcal{H}^{(1)}(k_1,\dots,k_s)} of strata calculated by A. Eskin and A. Okounkov. As an outcome of these works, it follows that the Siegel-Veech constants of Masur-Veech measures are rational numbers. In particular, by combining this fact with EKZ formula, we deduce that:

Corollary 7 (Eskin-Kontsevich-Zorich) For the Masur-Veech measures {\mu_{MV}}, one has

\displaystyle \lambda_1^{\mu_{MV}}+\dots+\lambda_g^{\mu_{MV}}\in\mathbb{Q}

Note that this corollary is quite spectacular because Lyapunov exponents normally are quantities coming from merely measurable subbundles and there is no a priori reason to believe in their rationality!

Actually, by looking at EKZ formula and taking the case of Masur-Veech measures as a prototype, it is tempting to conjecture that the sums of Lyapunov exponents of KZ cocycle with respect to {\textrm{SL}(2,{\mathbb R})}-invariant probabilities are always rational. In this direction, let us point out that the following result of Eskin-Kontsevich-Zorich allowing to compute (and show the rationality of) Siegel-Veech constants of measures coming from square-tiled surfaces.

Let {S_0=(M_0,\omega_0)} be a square-tiled surface, i.e., {S_0} comes from a finite covering {(M_0,\omega_0)\rightarrow\mathbb{T}^2=(\mathbb{C}/(\mathbb{Z}\oplus i\mathbb{Z}), dz)} branched only at {0}. Since {SL(2,\mathbb{Z})} is the stabilizer of {\mathbb{T}^2} in {SL(2,\mathbb{R})} (when the periods of {(M_0,\omega_0)} generate the lattice {\mathbb{Z}\oplus i\mathbb{Z}}), the {SL(2,\mathbb{Z})}-orbit of {(M_0, \omega_0)} give all square-tiled surfaces in the {SL(2,\mathbb{R})}-orbit of {(M_0,\omega_0)}. Moreover, since the Veech group {SL(M_0,\omega_0)} is a finite-index subgroup of {SL(2,\mathbb{Z})}, one has

\displaystyle SL(2,\mathbb{Z})\cdot (M_0,\omega_0)=\{S_0,S_1,\dots,S_{k-1}\},

where {k=[SL(2,\mathbb{Z}):SL(M_0,\omega_0)]=\#SL(2,\mathbb{Z})\cdot (M_0,\omega_0)}.

In this context, for each {S_j\in SL(2,\mathbb{Z})\cdot (M_0,\omega_0)}, we write {S_j=\bigcup C_{ij}} where {C_{ij}} are the maximal horizontal cylinders of {S_j}, and we denote the width and height of {C_{ij}} by {w_{ij}} and {h_{ij}}.

Theorem 8 (Eskin-Kontsevich-Zorich) The Siegel-Veech constant of the {SL(2,\mathbb{R})}-invariant {g_t}-ergodic probability {\mu} supported on the {SL(2,\mathbb{R})}-orbit of the square-tiled surface {(M_0,\omega_0)} is

\displaystyle \frac{1}{\# SL(2,\mathbb{Z})\cdot (M_0,\omega_0)}\sum\limits_{S_j\in SL(2,\mathbb{Z})\cdot (M_0,\omega_0)}\sum\limits_{S_j=\bigcup C_{ij}}\frac{h_{ij}}{w_{ij}}

In particular, the sum of Lyapunov exponents of such a measure satisfies

\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}\in\mathbb{Q}

Remark 19 Actually, the rationality result in the previous theorem is part of a slightly more general result of Eskin-Kontsevich-Zorich. Indeed, they showed that the Kontsevich-Forni formula implies that the sums of Lyapunov exponents of KZ cocycle with respect to measures supported on Teichmüller curves (i.e., closed {\textrm{SL}(2,{\mathbb R})}-orbits) are essentially given by the orbifold degree of the determinant line bundle of the Hodge bundle over these Teichmüller curves (a rational number as any orbifold degree of a line bundle). In particular, this generalizes the case of square-tiled surfaces because it is known that their {\textrm{SL}(2,{\mathbb R})}-orbits are closed.

For example, the picture below illustrates the computation of the {SL(2,\mathbb{Z})}-orbit of a {L}-shaped square-tiled surface {(M_0,\omega_0)} with {3} squares (shown in the middle of the picture):

Here, we’re using the fact that the group {SL(2,\mathbb{Z})} is generated by the matrices {S=\left(\begin{array}{cc}1&0\\1&1\end{array}\right)} and {T=\left(\begin{array}{cc}1&1\&1\end{array}\right)}, so that {SL(2,\mathbb{Z})}-orbits of square-tiled surfaces can be determined by successive applications of {S} and {T}.

From the picture we infer that {\#SL(2,\mathbb{Z})\cdot(M_0,\omega_0)=3} and

  • {(M_0,\omega_0)=C_{10}\cup C_{20}} where {C_{10}, C_{20}} are horizontal maximal cylinders with {h(C_{10})=h(C_{20})=1} and {w(C_{10})=1}, {w(C_{20})=2};
  • {(M_1,\omega_1):=S\cdot(M,\omega_0)=C_{11}} where {C_{11}} is a horizontal cylinder of heigth {1} and width {3};
  • {(M_2,\omega_2):=T\cdot(M_0,\omega_0) = C_{12}\cup C_{22}} where {C_{12}, C_{22}} are horizontal maximal cylinders with {h(C_{12})=h(C_{22})=1} and {w(C_{12})=1}, {w(C_{22})=2}

By plugging this into Theorem 8, we get that the Siegel-Veech constant of the {SL(2,\mathbb{R})}-invariant probability supported on {SL(2,\mathbb{R})\cdot(M_0,\omega_0)} is

\displaystyle \frac{1}{3}\left\{\left(\frac{1}{1}+\frac{1}{2}\right)+\frac{1}{3}+\left(\frac{1}{1}+\frac{1}{2}\right)\right\} = \frac{10}{9}

3.2. Some of consequences of EKZ formula

Let’s now apply Eskin-Kontsevich-Zorich formula to show the positivity of some Lyapunov exponents of KZ cocycle in the higher genus case:

Proposition 9 (Eskin-Kontsevich-Zorich) Let {\mu} be a ergodic {\textrm{SL}(2,{\mathbb R})}-invariant probability measure on a connected component {\mathcal{C}} of a stratum {\mathcal{H}^{(1)}(k_1,\dots,k_s)} of unit area Abelian differentials of genus {g\geq 7}. Then,

\displaystyle \lambda_2^{\mu}>0

(and, actually, {\lambda_{[(g-1)g/(6g-3)]+1}^{\mu}>0}).

Proof: Since {\lambda_1^{\mu}=1}, it suffices to show that the right-hand side of Eskin-Kontsevich-Zorich formula is {>1} to get that {\lambda_2^{\mu}>0}, and this follows from the computation

\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\frac{1}{12}\sum\limits_{j=1}^s\frac{k_j(k_j+2)}{(k_j+1)}+c(\mu)\geq \frac{1}{12}\sum\limits_{j=1}^s\frac{k_j(k_j+2)}{(k_j+1)}

\displaystyle >\frac{1}{12}\sum\limits_{j=1}^s k_j=\frac{2g-2}{12}\geq 1

based on the non-negativity of the Siegel-Veech constant {c(\mu)} and the assumption {g\geq 7}. \Box

Another interesting consequence of EKZ formula is the following. A. Eskin, M. Kontsevich and A. Zorich also showed in their article a version of the their formula for quadratic differentials, and they used it to compute Siegel-Veech constants of {SL(2,\mathbb{R})}-invariant {g_t}-ergodic probabilities {\mu} supported in the hyperelliptic connected components {\mathcal{H}_{hyp}(2g-2)} and {\mathcal{H}_{hyp}(g-1,g-1)} of the strata {\mathcal{H}(2g-2)} and {\mathcal{H}(g-1,g-1)}. The outcome of their computation is the fact that Siegel-Veech constant of any such {SL(2,\mathbb{R})}-invariant {g_t}-ergodic {\mu} is

\displaystyle c(\mu)=\left\{\begin{array}{cc}\frac{g(2g+1)}{3(2g-1)} & \textrm {if }\textrm{supp}(\mu)\subset \mathcal{H}_{hyp}(2g-2) \\ \frac{2g^2+3g+1}{6g} & \textrm {if }\textrm{supp}(\mu)\subset \mathcal{H}_{hyp}(2g-2)\end{array}\right.

Hence, by combining this with EKZ formula, we conclude that

Theorem 10 The sum of Lyapunov exponents is

\displaystyle \lambda_1^{\mu}+\dots+\lambda_g^{\mu}=\left\{\begin{array}{cc}\frac{g^2}{(2g-1)} & \textrm {if }\textrm{supp}(\mu)\subset \mathcal{H}_{hyp}(2g-2) \\ \frac{g+1}{2} & \textrm {if }\textrm{supp}(\mu)\subset \mathcal{H}_{hyp}(g-1,g-1)\end{array}\right.

In particular, since the sole two strata {\mathcal{H}(2)} and {\mathcal{H}(1,1)} in genus {2} are hyperelliptic connected components, one has that, for any {SL(2,\mathbb{R})}-invariant {g_t}-ergodic {\mu},

\displaystyle \lambda_2^{\mu}=\left\{\begin{array}{cc}1/3 & \textrm {if }\textrm{supp}(\mu)\subset \mathcal{H}(2) \\ 1/2 & \textrm {if }\textrm{supp}(\mu)\subset \mathcal{H}(1,1)\end{array}\right.

because {\lambda_1^{\mu}=1}. This fact was conjectured by M. Kontsevich and A. Zorich, and it was firstly demonstrated by M. Bainbridge a few years before the Eskin-Kontsevich-Zorich article was available.

Remark 20 In a very recent work, D. Aulicino further studied the problem of classifying {SL(2,\mathbb{R})}-invariant measures with totally degenerate spectrum from the point of view of the Teichmüller disks (i.e., {SL(2,\mathbb{R})}-orbits) contained in the rank-one locus. More precisely, following G. Forni, we define the rank-{k} locus of the moduli space {\mathcal{H}_g} of Abelian differentials of genus {g} is {\mathcal{D}_g(k):=\{\omega\in\mathcal{H}_g: \textrm{rank}(B_{\omega}) \leq k\}}. Note that {\mathcal{D}_g(1)\subset\dots\subset\mathcal{D}_g(g-1)}. In the literature, the locus {\mathcal{D}_g(g-1)} is sometimes called determinant locus (because {\mathcal{D}_g(g-1)=\{\omega\in\mathcal{H}_g: \det B_{\omega}=0\}}). Observe that these loci are naturally related to the study of Lyapunov exponents of KZ cocycle: for instance, by Theorem 5, any {SL(2,\mathbb{R})}-invariant probability {\mu} with {\textrm{supp}(\mu)\subset\mathcal{D}_g(1)} has totally degenerate spectrum. In his work, D. Aulicino showed that there are no Teichmüller disks {SL(2,\mathbb{R})\cdot(M,\omega)} contained in {\mathcal{D}_g(1)} for {g=2} or {g\geq 13}, the so-called Eierlegende Wollmilchsau and Ornithorynque are the sole Teichmüller disks contained in {\mathcal{D}_3(1)} and {\mathcal{D}_4(1)}, and, furthermore, if there are no Teichmüller curves contained in {\mathcal{D}_5(1)}, then there are no Teichmüller disks contained in {\mathcal{D}_g(1)} for {g\geq 5}. It is worth to point out that Teichmüller disks are more general objects than regular affine measures, so that Proposition 9 doesn’t allow to recover the results of D. Aulicino.

Remark 21 The Eierlegende Wollmilchsau and Ornithorynque are two square-tiled surfaces with the following properties:

  • Eierlegende Wollmilchsau lives in {\mathcal{H}(1,1,1,1)} and it is decomposed in two maximal horizontal cylinders {C_1}, {C_2} with {h(C_1)=h(C_2)=1} and {w(C_1)=w(C_2)=4}, and {SL(2,\mathbb{Z})}-orbit is a singleton;
  • Ornithorynque lives in {\mathcal{H}(2,2,2)} and it is decomposed in two maximal horizontal cylinders {C_1}, {C_2} with {h(C_1)=h(C_2)=1} and {w(C_1)=w(C_2)=6} and its {SL(2,\mathbb{Z})}-orbit is a singleton.

By plugging these facts into Theorem 8, one can compute the Siegel-Veech constants of the measures {\mu_{\mathcal{EW}}} and {\mu_{\mathcal{O}}} associated to these examples, and then, by EKZ formula, one can calculate the sum of their Lyapunov exponents. By doing so, one finds:

\displaystyle \lambda_1^{\mu_{\mathcal{EW}}}+\lambda_2^{\mu_{\mathcal{EW}}}+\lambda_3^{\mu_{\mathcal{EW}}} = \frac{1}{12}\cdot4\cdot \frac{1\cdot 3}{2} + \frac{1}{1}\cdot \left(\frac{1}{4}+\frac{1}{4}\right)=1

and

\displaystyle \lambda_1^{\mathcal{O}}+\lambda_2^{\mathcal{O}}+\lambda_3^{\mathcal{O}}+\lambda_4^{\mathcal{O}} = \frac{1}{12}\cdot 3\cdot \frac{2\cdot 4}{3} + \frac{1}{1}\cdot \left(\frac{1}{6}+\frac{1}{6}\right)=1

Since {\lambda_1^{\mu}=1} for any {g_t}-invariant ergodic {\mu}, one concludes that {\lambda_2^{\mu_{\mathcal{EW}}}=\lambda_2^{\mu_{\mathcal{EW}}}=0} and {\lambda_2^{\mu_{\mathcal{O}}}=\lambda_3^{\mu_{\mathcal{O}}}=\lambda_4^{\mu_{\mathcal{O}}}=0}, a fact that known (by other methods) from results of Forni and his collaborator.

Remark 22 In the case of {\mu} coming from square-tiled surfaces {(M_0,\omega_0)}, the formula in Theorem 8 for {c(\mu)} combined with EKZ formula suggests that one can write down computer programsto calculate the sum of Lyapunov exponents. Indeed, firstly we observe that square-tiled surfaces are intimately related to pairs of permutations {h,v\in S_N}: a pair of permutations {h,v\in S_N} gives rise to a square-tiled surface with {N} squares by taking {N} unit squares {Q_i}, {i=1,\dots, N}, and by gluing (by translations) the rightmost vertical side of {Q_i} to the leftmost vertical side of {Q_{h(i)}} and the topmost horizontal side of {Q_i} to the bottommost horizontal side of {Q_{v(i)}}. Of course, by renumbering the squares of a given square-tiled surface we may end up with different pairs of permutations, so that a square-tiled surface determines a pair {h,v\in S_N} modulo simultaneous conjugation, i.e., modulo the equivalence relation {(h',v')\sim(h,v)} if and only if {h'=\phi h \phi^{-1}} and {v'=\phi v\phi^{-1}} for some {\phi\in S_N}. Finally, it is possible to check that the action of the matrices {S=\left(\begin{array}{cc}1 & 0 \\ 1 & 1\end{array}\right)} and {T=\left(\begin{array}{cc}1 & 1 \\ 0 & 1\end{array}\right)} on square-tiled surfaces translation into the action {S(h,v)=(hv^{-1},v)} and {T(h,v)=(h,vh^{-1})} on pairs of permutations. Secondly, in this language, the heights and widths of its horizontal cylinders are determined by the cycles of the permutation {h}. So, by recalling that {S} and {T} generate {SL(2,\mathbb{Z})}, we can use the right-hand side of the formula in Theorem 8 to convert the computation of the Siegel-Veech constant of the {SL(2,\mathbb{R})}-invariant probability supported on {(M_0,\omega_0)} into a combinatorial calculation with pairs of permutations that an adequate computer program can perform.

In fact, such computer programs for Mathematica and SAGE were written by, e.g., A. Zorich and V. Delecroix, and it is likely that they will be publicly available soon.


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