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).

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