SPCS 4

Posted by: matheuscmss | February 3, 2012

SPCS 4

As usual, we’ll start with a quick revision of SPCS 3. Let $M$ be an origami and denote by $\Gamma=Aut(M)$ the (finite) group of its automorphisms. Given $a\in \textrm{Irr}_{\mathbb{R}}(\Gamma)$ a $\Gamma$ -irreducible $\mathbb{R}$ -representation, let $V_a$ be the $\Gamma$ -module associated to $a$ . In this way, we wrote the interesting part $H_1^{(0)}(M,\mathbb{R})$ of the (first absolute) homology of $M$ as

$H_1^{(0)}(M,\mathbb{R})=\bigoplus\limits_{a\in\textrm{Irr}_{\mathbb{R}}(\Gamma)} W_a$

where $W_a\simeq V_a^{\ell_a}$ is the isotypical component associated to $V_a$ . Then, we saw that $a$ could have three types (real, complex or quaternionic), and we studied the nature of the group $Sp(W_a)$ of automorphisms of $W_a$ (as $\mathbb{R}(\Gamma)$ -module) preserving the restriction to $W_a$ of the natural (symplectic) intersection form on $H_1(M,\mathbb{R})$ . This is illustrated in the following table:

${a}$	${\textit{real}}$	${\textit{complex}}$	${\textit{quaternionic}}$
${\textrm{dim}_{\mathbb{R}}(V_a)}$	${\textrm{dim}_{\mathbb{R}}(V_a)}$	${2\textrm{dim}_{\mathbb{C}}(V_a)}$	${4\textrm{dim}_{\mathbb{H}}(V_a)}$
${\ell_a=\frac{\textrm{dim}_{\mathbb{R}}(W_a)}{\textrm{dim}_{\mathbb{R}}(W_a)}}$	${\ell_a}$ even	${\ell_a\neq 1}$	${\ell_a=\frac{1}{2}\ell_{\alpha}}$
${Sp(W_a)}$	${Sp(\ell_a,\mathbb{R})}$	${U_{\mathbb{C}}(p,q)}$	${U_{\mathbb{H}}(p,q)}$

Here, $Sp(\ell_a,\mathbb{R})$ is the usual symplectic group over $\mathbb{R}$ , while $U_{\mathbb{C}}(p,q)$ and $U_{\mathbb{H}}(p,q)$ resp. are the unitary group of matrices (with coefficients on $\mathbb{C}$ and $\mathbb{H}$ resp.) preserving the Hermitian form

$\sum\limits_{m=1}^p \overline{z}_m z_m - \sum\limits_{m=p+1}^{p+q} \overline{z}_m z_m$

where $p+q=\ell_a$ . Also, in the case of $a$ quaternionic, $\ell_{\alpha}$ is the multiplicity of the unique complex representation $\alpha$ with $a=\{\alpha\}$ .

After this quick revision, we are ready to discuss (below the fold) the content of J.-C. Yoccoz’s 4th lecture, that is, how to use the information above on the structure of the group $Sp(W_a)$ can be used to discuss Lyapunov exponents of the Kontsevich-Zorich cocycle on the $SL(2,\mathbb{R})$ -orbit of origamis.

-Moduli spaces of translation surfaces-

Let $M$ be a topological surface, and let $\Sigma\subset M$ a finite subset of points, say $\Sigma=\{A_1,\dots, A_s\}$ , $s\geq 1$ . Consider $\kappa=(k_1,\dots,k_s)$ a list of integers $k_i\geq 1$ such that

$\sum\limits_{i=1}^s (k_i-1)=2g-2$

Here, we will think that $k_i$ is “attached” to $A_i$ , and sometimes we denote $k_i=:k_{A_i}$ .

Let $\textrm{Diff}(M,\Sigma,\kappa)$ be the group of homeomorphisms $f$ of $M$ preserving $\Sigma$ and $\kappa$ , that is, $k_{f(A_i)}=k_{A_i}$ for each $1\leq i\leq s$ , and let $\textrm{Diff}_0(M,\Sigma,\kappa)$ be the connected component of the identity element in $\textrm{Diff}(M,\Sigma,\kappa)$ . The quotient group $\textrm{Mod}(M,\Sigma,\kappa):=\textrm{Diff}(M,\Sigma,\kappa)/\textrm{Diff}_0(M,\Sigma,\kappa)$ is the so-called modular group.

We denote by $\mathcal{T}(M,\Sigma,\kappa)$ the set of translation structures on $(M,\Sigma,\kappa)$ , and we introduce

$\mathcal{Q}(M,\Sigma,\kappa)=\mathcal{T}(M,\Sigma,\kappa)/\textrm{Diff}_0(M,\Sigma,\kappa)$

the Teichmuller space of translation structures on $(M,\Sigma,\kappa)$ , and

$\mathcal{M}(M,\Sigma,\kappa)=\mathcal{T}(M,\Sigma,\kappa)/\textrm{Diff}(M,\Sigma,\kappa) = \mathcal{Q}(M,\Sigma,\kappa)/\textrm{Mod}(M,\Sigma,\kappa)$

the moduli space of translation structures on $(M,\Sigma,\kappa)$ .

-Action of $GL(2,\mathbb{R})$ on $\mathcal{T}, \mathcal{Q}$ and $\mathcal{M}$ –

Given $g\in GL(2,\mathbb{R})$ , and $\zeta\in\mathcal{T}(M,\Sigma,\kappa)$ a translation structure associated to a translation atlas $(\varphi_{\alpha})$ on $M-\Sigma$ , we define $g_*\zeta$ the translation structure associated to the translation atlas $g\circ\varphi_{\alpha}$ . This gives a $GL(2,\mathbb{R})$ -action on $\mathcal{T}(M,\Sigma,\kappa)$ commuting with the action of $\textrm{Diff}(M,\Sigma,\kappa)$ (as $GL(2,\mathbb{R})$ acts by post-composition with translation charts while $\textrm{Diff}(M,\Sigma,\kappa)$ acts by pre-composition with translation charts). Hence, it induces $GL(2,\mathbb{R})$ -actions on $\mathcal{Q}(M,\Sigma,\kappa)$ and $\mathcal{M}(M,\Sigma,\kappa)$ .

In this language, the Teichmuller geodesic flow $T^t$ is given by the action of the diagonal matrices $\left(\begin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array}\right)$ .

-Veech group-

Given $[\zeta]\in\mathcal{M}(M,\Sigma,\kappa)$ , its Veech group $GL(\zeta)$ is the stabilizer of $[\zeta]$ under the action of $GL(2,\mathbb{R})$ . Observe that $g\in GL(\zeta)$ implies $\det g=\pm1$ (as our translation surfaces have finite area). This makes it natural to introduce $SL(\zeta):=GL(\zeta)\cap SL(2,\mathbb{R})$ .

-Affine group-

Given a translation structure $\zeta$ , we denote by

$\textrm{Aff}(\zeta):=\{f\in \textrm{Diff}(M,\Sigma,\kappa): f \textrm{ is locally affine when read}$

$\textrm{ in the charts of } \zeta\}$

the affine group of $\zeta$ .

Note that we have a well-defined map $\textrm{Aff}(\zeta)\ni f\mapsto Df\in GL(2,\mathbb{R})$ obtained by looking at the linear part $Df$ of $f$ in the charts of $\zeta$ . This map is part of an exact sequence

$1\to \textrm{Aut}(\zeta)\to\textrm{Aff}(\zeta)\stackrel{D}{\to} GL(\zeta)\to 1$

Observe also that $A\in\textrm{Aff}(\zeta)$ induces an isomorphism $(M,g_*\zeta)\stackrel{\simeq}{\to} (M,\zeta)$ where $g=DA$ .

Finally, it is possible to check that the natural map $Aff(\zeta)\to \textrm{Mod}(M,\Sigma,\kappa)$ is injective and its image is exactly the set of $\varphi\in\textrm{Mod}(M,\Sigma,\kappa)$ such that $\varphi_*([\zeta])\in GL(2,\mathbb{R})\cdot[\zeta]$ .

-Veech surfaces-

Definition. $(M,\zeta)$ is a Veech surface whenever $SL(\zeta)$ is a lattice in $SL(2,\mathbb{R})$ .

Example. Let $(M,\zeta)$ be a reduced origami (i.e., whose periods generate $\mathbb{Z}^2$ ). Then, $GL(\zeta)$ is a finite index subgroup of $GL(2,\mathbb{Z})$ , and hence any origami is a Veech surface.

By a result of J. Smillie, $SL(2,\mathbb{R})\cdot[\zeta]\subset\mathcal{M}(M,\Sigma,\kappa)$ is closed if and only if $(M,\zeta)$ is a Veech surface.

Observe that, in the case of Veech surfaces, $SL(2,\mathbb{R})\cdot[\zeta]$ is naturally identified with the unit tangent bundle of the hyperbolic surface $\mathcal{H}/SL(\zeta)$ . Here, $\mathcal{H}$ denotes Poincare’s upper-half plane (we avoided the more standard notation $\mathbb{H}$ as we’re currently using this symbol for Hamilton’s quaternions). In the literature, $\mathcal{H}/SL(\zeta)$ is called Teichmuller surface (or Teichmuller curve).

In this language, the restriction of the Teichmuller flow $T^t$ to $SL(2,\mathbb{R})\cdot[\zeta]$ is the geodesic flow on $\mathcal{H}/SL(\zeta)$ .

-Kontsevich-Zorich cocycle-

We begin with the following trivial bundle over $\mathcal{Q}(M,\Sigma,\kappa)$ (whose fibers are $2g$ -dimensional):

$\mathcal{Q}(M,\Sigma,\kappa)\times H_1(M,\mathbb{R})$

Observe that the modular group $\textrm{Mod}(M,\Sigma,\kappa)$ acts on both factors of the total space of this trivial bundle: the action on $\mathcal{Q}(M,\Sigma,\kappa)$ was already discussed, while the action on the fibers $H_1(M,\mathbb{R})$ is by pull-back (as usual). In particular, by using this action we can consider the quotient bundle

$(\mathcal{Q}(M,\Sigma,\kappa)\times H_1(M,\mathbb{R}))/\textrm{Mod}(M,\Sigma,\kappa)$

over $\mathcal{M}(M,\Sigma,\kappa)$ . This quotient bundle is a non-trivial bundle known in the literature as Hodge bundle. The Hodge bundle is the phase space of a cocycle (in the sense of Dynamical Systems)

$G_{KZ}^t([\zeta],v)=(T^t[\zeta],v)$

called Kontsevich-Zorich cocycle. At first sight this cocycle seems to be trivial, but this is far from being true because the Hodge bundle itself is non-trivial!

Observe that $G_{KZ}^t$ is a symplectic cocycle (as the natural intersection form on $H_1(M,\mathbb{R})$ is preserved).

Furthermore, this cocycle admits plenty of invariant measures. Below we review the construction of some of them.

a. Masur-Veech measures

The so-called period map $\mathcal{Q}(M,\Sigma,\kappa)\stackrel{\Theta}{\to}\textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{C})$ given by

$\Theta(\zeta)(\gamma)=\int_\gamma\omega \quad \textrm{ for }\gamma\in H_1(M,\Sigma,\mathbb{Z})$

(where $\omega$ is the natural holomorphic 1-form associated to $\zeta$ by pull-back $dz$ from the plane through the translation charts) is a local homeomorphism. Since $\textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{C})$ is a complex vector space of dimension $2g+s-1$ containing a canonical lattice $\textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{Z}\oplus i\mathbb{Z})$ (the image under $\Theta$ of origamis), we can induce a Lebesgue measure on $\mathcal{M}(M,\Sigma,\kappa)$ by using $\Theta$ to pull-back the Lebesgue measure on $\textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{C})$ normalized so that the canonical lattice $\textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{Z}\oplus i\mathbb{Z})$ has covolume 1.

This Lebesgue measure on $\mathcal{M}(M,\Sigma,\kappa)$ is $SL(2,\mathbb{R})$ -invariant, but it has infinite mass. On the other hand, the induced Lebesgue measure $\mu_{MV}$ on the moduli space $\mathcal{M}_1(M,\Sigma,\kappa)$ of translation structures leading to unit area surfaces has better properties (as it was shown by H. Masur and W. Veech independently):

Theorem (H. Masur, W. Veech). The measure $\mu_{MV}$ has finite mass. The Teichmuller flow is ergodic (and mixing) on each connected component of $\mathcal{M}_1(M,\Sigma,\kappa)$ .

In the literature, the $SL(2,\mathbb{R})$ -invariant measure $\mu_{MV}$ is called Masur-Veech measure. Of course, the Masur-Veech measure is invariant under the action of $SL(2,\mathbb{R})$ , it is Teichmuller flow invariant (in particular). So, this provides a class of Teichmuller flow invariant measures.

b. Haar measures on Teichmuller surfaces

Let $\zeta$ be a Veech surface. The unit tangent bundle $SL(2,\mathbb{R})/SL(\zeta)$ of the Teichmuller surface $\mathcal{H}/SL(\zeta)$ has a natural finite and $SL(2,\mathbb{R})$ -invariant Haar measure. The Teichmuller flow is ergodic (and mixing) with respect to this measure because we are in the setting of geodesic flows on the unit tangent bundle of a negatively curved surface.

-Oseledets theorem-

Once we fix an ergodic Teichmuller flow invariant finite measure $\mu$ , we can start the statistical study of the Kontsevich-Zorich cocycle with the aid of Oseledets theorem.

Roughly speaking, Oseledets theorem ensures (in our setting) the existence of real numbers $\theta_1>\dots>\theta_r$ (Lyapunov exponents) such that for $\mu$ -almost every $[\zeta]$ one has a decomposition

$H_1(M,\mathbb{R})=\bigoplus E(\theta_i,[\zeta])=\bigoplus E(\theta_i)$

where

$E(\theta_i)=E(\theta_i,[\zeta])=\{v\in H_1(M,\mathbb{R}): \lim\limits_{t\to\pm\infty}\frac{\log\|p_2(G_{KZ}^t([\zeta],v))\|}{t}=\theta_i\}$

(and $p_2$ denotes the projection into the fibers, i.e., second factor, of Hodge bundle).

Here $\|.\|$ is any norm on the fibers of the Hodge bundle such that $G_{KZ}^t$ verifies the following integrability condition: $\log\|G_{KZ}^1\|, \log\|G_{KZ}^{-1}\|\in L^1(\mu)$ . In general this integrability condition is a crucial issue for the application of Oseledets theorem, but in the context of the Kontsevich-Zorich cocycle it is possible to prove that it is automatically satisfied: for instance, by the works of G. Forni, $\log\|G_{KZ}^1\|, \log\|G_{KZ}^{-1}\|\leq 1$ for a particular norm known as the Hodge norm, so that the integrability conditions holds (with respect to the Hodge norm) as soon as $\mu$ has finite mass.

By the symplecticity of $G_{KZ}^t$ one can show that $\theta_i=-\theta_{1+r-i}$ and $\textrm{dim}(E(\theta_i)) = \textrm{dim}(E(\theta_{1+r-i}))$ .

Remark. It is not hard to check from definitions that $\theta_1=1$ and $\theta_r=-1$ .

It was shown by W. Veech that $\textrm{dim}(E(\theta_1))=1$ in general.

On the other hand, for the Lebesgue/Masur-Veech measure $\mu_{MV}$ on a connected component of $\mathcal{M}_1(M,\Sigma,\kappa)$ , G. Forni proved that $G_{KZ}^t$ is hyperbolic in the sense that $\theta_i\neq 0$ for all $i$ , and A. Avila and M. Viana improved this result (and solved the so-called Kontsevich-Zorich conjecture) by proving that $\textrm{dim}E(\theta_i)=1$ for all $i$ .

More recently, A. Eskin, M. Kontsevich and A. Zorich developed explicit formulas for the sum of the non-negative Lyapunov exponents with respect to any “good ” $SL(2,\mathbb{R})$ -invariant finite measure.

Our discussion so far concerned the Kontsevich-Zorich cocycle and its Lyapunov exponents with respect to general Teichmuller flow/ $SL(2,\mathbb{R})$ invariant ergodic measures. From now on, we will specialize to the case of origamis.

-Lyapunov exponents associated to origamis-

Let $(M,\zeta)$ be an origami. The decomposition

$H_1(M,\mathbb{R}) = H_1^{st}(M,\mathbb{R})\oplus H_1^{(0)}(M,\mathbb{R})$

is:

constant along $SL(2,\mathbb{R})\cdot[\zeta]$ ;
invariant under the action of the affine group $\textrm{Aff}(\zeta)$ ;
preserved by $G_{KZ}^t$ .

The factor $H_1^{st}(M,\mathbb{R})$ corresponds to the Lyapunov exponents $\theta_1=1$ and $\theta_r=-1$ , and the action of the group $\textrm{Aff}^+(\zeta)$ of orientation-preserving elements of $\textrm{Aff}(\zeta)$ factors through $\textrm{Aff}^+(\zeta)\to SL(\zeta)\subset SL(2,\mathbb{R})$ where $SL(2,\mathbb{R})$ acts in the standard way on $\mathbb{R}^2\simeq H_1(\mathbb{T}^2,\mathbb{R})\simeq H_1^{st}(M,\mathbb{R})$ . This justifies the superscript “st” on $H_1^{st}(M,\mathbb{R})$ as it is the factor of homology where we act in a standard way.

Next, we recall the notation $\Gamma=Aut(\zeta)(\simeq N/H)$ from previous lectures. Given $A\in \textrm{Aff}(\zeta)$ , we define an automorphism $\rho_A$ of $\Gamma$ by

$\rho_A(\gamma)=A\circ\gamma\circ A^{-1}$

In this way, we obtain a homomorphism $\textrm{Aff}(\zeta)\stackrel{\rho}{\to}\textrm{Aut}(\Gamma)$ . Since $\Gamma$ and, a fortiori, $\textrm{Aut}(\Gamma)$ are finite, we have that

$\textrm{Aff}_*(\zeta) := \textrm{Ker}(\rho)$

is a finite-index subgroup of $\textrm{Aff}(\zeta)$ .

By definition, any $A\in\textrm{Aff}_*(\zeta)$ commutes with the action of $\Gamma$ , so that the induced map

$A_*: H_1^{(0)}(M,\mathbb{R})\to H_1^{(0)}(M,\mathbb{R})$

is a homomorphism of $\Gamma$ -modules preserving the natural intersection form on $H_1^{(0)}(M,\mathbb{R})$ .

Consider the decomposition

$(1) \quad H_1^{(0)}(M,\mathbb{R}) = \bigoplus\limits_{a\in \textrm{Irr}_{\mathbb{R}}(\Gamma)} W_a$

into isotypical irreducible representations.

It follows that, for $A\in\textrm{Aff}_*(\zeta)$ , one has $A_*(W_a)=W_a$ , and thus $A_*|_{W_a}\in Sp(W_a)$ . In other words, we have a natural homomorphism

$\textrm{Aff}_*(\zeta)\to\prod\limits_a Sp(W_a)$

The decomposition (1) above is constant along the $SL(2,\mathbb{R})$ orbit of $\zeta$ , so that it is invariant by $G_{KZ}^t$ (as $\textrm{Aut}(g_*\zeta)=g\textrm{Aut}(\zeta)g^{-1}$ and any character is invariant under conjugation). This allows us to fix once and for all $a\in\textrm{Irr}_{\mathbb{R}}(\Gamma)$ , and consider $G_{KZ}^t|_{W_a}$ and its Lyapunov exponents denoted by $(1>)\theta_1>\dots>\theta_r (>-1)$ by a slight abuse of notation.

Let $[\zeta_*]\in SL(2,\mathbb{R})\cdot[\zeta]$ be a generic point (in the sense that Oseledets theorem can be applied to it). We can write

$W_a=\bigoplus E(\theta_i)=E(\theta_i,[\zeta_*])$

Since $\textrm{Aff}_*(\zeta)$ is a finite index subgroup of $\textrm{Aff}(\zeta)$ and a generic point $[\zeta_*]$ is recurrent, we can take a sequence of times $t_n\to\pm\infty$ as $n\to\pm\infty$ such that $T^{t_n}([\zeta_*])$ is very close to $[\zeta_*]$ in such a way that by “closing up” the segment $\{T^s([\zeta_*]): 0\leq s\leq t_n\}$ with a bounded path between $T^{t_n}([\zeta_*])$ and $[\zeta_*]$ , the corresponding closed path leads to an affine diffeomorphism $A_n\in\textrm{Aff}_*(\zeta)$ . See the figure below.

In this way, since $\|A_n\|\sim e^{|t_n|}$ (as the largest exponent of the Kontsevich-Zorich cocycle is $1$ ) we have

$E(\theta_i)=\{v\in W_a: \lim\limits_{n\to\pm\infty}\frac{\log\|A_n v\|}{\log\|A_n\|}=\pm\theta_i\}$

Proposition. $E(\theta_i)$ is a $\Gamma$ -module.

Proof. Because $\Gamma$ is a finite group (and the definition of Lyapunov exponents doesn’t depend on the norm), we may assume that the norm $\|.\|$ is $\Gamma$ -invariant (by the usual averaging argument). Since $A_n\in\textrm{Aff}_*(\zeta)$ , it commutes with the action $\Gamma$ . It follows that, for any $\gamma\in\Gamma$ ,

$\log\|A_n\gamma v\|=\log\|\gamma A_nv\|=\log\|A_n v\|$

and thus $\gamma v\in E(\theta_i)$ whenever $v\in E(\theta_i)$ . $\square$

This proposition implies that $E(\theta_i)\simeq V_a^{\ell(\theta_i)}$ . Furthermore,

$\sum\limits_{i=1}^r\ell(\theta_i)=\ell_a$ (as $W_a\simeq V_a^{\ell_a}$ ), and
$\ell(\theta_{1+r-i})=\ell(-\theta_i)=\ell(\theta_i)$ (by symplecticity).

It is worth to note that to the best of our knowledge, the following question is open:

Question. Is there an example where $\ell(\theta_i)>1$ for some $\theta_i\neq0$ .

In general, we have that $\textrm{dim}_{\mathbb{R}}(E(\theta_i)) = \ell(\theta_i)\cdot\textrm{dim}_{\mathbb{R}}(V_a)$ , and

$\textrm{dim}_{\mathbb{R}}(V_a) = 2 \textrm{dim}_{\mathbb{C}}(V_a)$ when $a$ is complex;
$\textrm{dim}_{\mathbb{R}}(V_a) = 4 \textrm{dim}_{\mathbb{H}}(V_a)$ when $a$ is quaternionic.

In fact, while this general discussion is the best one can do concerning the case $a$ real, it is possible to say a little more about Lyapunov exponents in the complex and quaternionic cases. This is the content of the last section of today’s post.

-Lyapunov exponents when $a$ is complex or quaternionic-

Recall that $Sp(W_a)\simeq U_{\mathbb{C}}(p,q)$ or $U_{\mathbb{H}}(p,q)$ when $a$ is complex or quaternionic.

Lemma 1. Let $v\in E(\theta_i), v'\in E(\theta_j)$ , where $\theta_i+\theta_j\neq 0$ . Then, $\langle v,v'\rangle_{W_a}=0$ , where $\langle .,.\rangle_{W_a}$ is the Hermitian form invariant by $Sp(W_a)$ of signature $(\overline{p}, \overline{q})$ with $\overline{p}:=\textrm{dim}_{\mathbb{C}/\mathbb{H}}V_a\cdot p$ and $\overline{q}:=\textrm{dim}_{\mathbb{C}/\mathbb{H}}V_a\cdot q$ . The real part of $\langle .,.\rangle_{W_a}$ considered as a real quadratic form has signature $(\textrm{dim}_{\mathbb{R}}V_a\cdot p, \textrm{dim}_{\mathbb{R}}V_a\cdot q)$ .

Proof. Without loss of generality we may assume $\theta_i+\theta_j<0$ (as the other case is symmetric). This implies that

$\langle v,v'\rangle_{W_a}=\langle A_n v,A_n v' \rangle_{W_a}\sim \|A_n\|^{\theta_i}\cdot\|A_n\|^{\theta_j}\stackrel{n\to\infty}{\to}0$

so that the proof is complete. $\square$

Lemma 2. Let $Q$ be a non-degenerate quadratic form of signature $(\overline{p},\overline{q})$ and let $E$ be a vector subspace such that $Q|_{E}=0$ . Then, $\textrm{dim}(E)\leq \min\{p,q\}$ .

Proof. Without loss of generality we may assume $\overline{p}\leq\overline{q}$ (as the other case is symmetric), and by appropriately choosing a basis,

$Q(x)=\sum\limits_{m=1}^{\overline{p}} x_m^2 - \sum\limits_{m=\overline{p}+1}^{\overline{p}+\overline{q}} x_m^2$

If $\textrm{dim}(E)\geq \overline{p}+1$ , we notice that there exists $v\in E-\{0\}$ such that $v_1=\dots=v_{\overline{p}}=0$ : indeed, the algebraic operation of annihilating a given coordinate reduces the dimension of $E$ by $1$ at most, so that we can find a non-trivial vector whose first $\overline{p}$ coordinates vanish inside any subspace of dimension $\overline{p}+1$ (at least). By the expression of $Q$ above, this would mean that $Q(v)<0$ , a contradiction with the assumption $Q|_{E}=0$ . $\square$

In the setting of origamis, we take $Q=\textrm{Re}\langle.,.\rangle_{W_a}$ and

$E^+:=\bigoplus\limits_{\theta_i>0}E(\theta_i)$

By Lemma 1, $Q|_{E^+}=0$ , so that, by Lemma 2, we have the following corollary

Corollary. $\textrm{dim}_{\mathbb{R}}(V_a)\cdot\sum\limits_{\theta_i>0}\ell(\theta_i):=\textrm{dim}E^+\leq \min\{\overline{p},\overline{q}\}$ .

Of course, a similar statement is true for

$E^-:=\bigoplus\limits_{\theta_j<0}E(\theta_j)$

In particular, we conclude the following estimate concerning zero Lyapunov exponents:

$\ell(0)\geq |p-q|$

Partly motivated by this discussion, we have the following questions:

Question. Is there an example where the inequality $\ell(0)\geq |p-q|$ is strict?

Question. When $a$ is real, is there an example with some zero Lyapunov exponent in $W_a$ ?

Question. In the case of hyperelliptic origamis (i.e., origamis whose underlying Riemann surface is hyperelliptic), is there a difference in our discussions so far if we replace $\Gamma=\textrm{Aut}(\zeta)$ by

$\Gamma=\widetilde{\textrm{Aut}}(\zeta)=\{A\in\textrm{Aff}(\zeta): DA=\pm \textrm{id}\}$ ?

Closing today’s post, we mention that the plan for the next lecture is to apply the discussions of SPCS 1, 2, 3 and 4 to concrete examples (especially regular and quasi-regular origamis).

Responses

[…] Carlos Matheus: Special curves in Hilbert modular surfaces and Lyapunov exponents of Prym eigenforms, SPCS 4 […]
By: Fourteenth Linkfest on February 7, 2012
at 5:58 pm

Reply

M	T	W	T	F	S	S
« Jan				Mar »
		1	2	3	4	5
6	7	8	9	10	11	12
13	14	15	16	17	18	19
20	21	22	23	24	25	26
27	28	29

	Felipe on Examples of Rauzy classes (aft…
	matheuscmss on Examples of Rauzy classes (aft…
	Felipe on Examples of Rauzy classes (aft…
	matheuscmss on On the continuity of Lyapunov…
	Reza on On the continuity of Lyapunov…

Disquisitiones Mathematicae

SPCS 4

Like this:

Related

Responses

Leave a Reply Cancel reply

Categories

Archives

Mathematics

politics

Recent Posts

Top Posts

Recent Comments