Posted by: matheuscmss | October 2, 2008

## A Teichmüller disk in genus 4 with degenerate Kontsevich-Zorich spectrum

Yesterday, I gave a talk on the “Seminaire de Theorie Ergodique” at the Universite de Paris 13 (Villetaneuse) about a short note entitled “An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum” that G. Forni and me just posted on the arXiv.

The Teichmuller geodesic flow acts (as any respectable geodesic flow) on the unit cotangent bundle of the Teichmuller space by parallel transport of points and vectors along geodesics of the Teichmuller metric (here one should be careful because the Teichmuller metric is not a Riemannian metric but only a Finsler metric).

Using the results of Alhfors and Bers in Riemann surface theory, it is possible to identify the unit cotangent bundle of the Teichmuller space with the space $Q_g^{(1)}$ of holomorphic quadratic differentials on a Riemann surface $M$ of genus $g$ with area 1 modulo the group $Diff^+_0(M)$ of orientation-preserving diffeomorphisms of $M$ which are isotopic to the identity. Note that $SL(2,\mathbb{R})$ acts naturally on $Q_g^{(1)}$ by linear transformations on the pairs $(\Re(q^{1/2}), \Im(q^{1/2}))$ of Abelian differentials. The interesting consequence of this point of view is that it turns out that the Teichmuller geodesic flow $G_t$ on $Q_g^{(1)}$ is simply the action of the diagonal subgroup $\textrm{diag}(e^t,e^{-t})$ on $Q_g^{(1)}$.

Now let me make a few general remarks about the fine structure of $Q_g^{(1)}$:

• $Q_g$ is stratified into analytic spaces $Q_{\kappa}^{(1)}$ obtained by fixing the multiplicities $\kappa=(k_1,\dots,k_{\sigma})$ of the zeroes $\Sigma_\kappa=\{p_1,\dots,p_{\sigma}\}$ of the quadratic differentials (here $\sum k_i=4g-4$ in view of the Riemann-Hurwitz theorem);
• $Q_g^{(1)}$ can be endowed with a natural notion of ‘Lebesgue measure’: for each stratum $Q_\kappa^{(1)}$ there is a $SL(2,\mathbb{R})$-invariant probability measure $\mu_\kappa^{(1)}$ in the same class of the Lebesgue measure on the local charts $Q_\kappa^{(1)}\to H^1(M,\Sigma_\kappa,\mathbb{C})$ given by the period map;

Once we know that $Q_g^{(1)}$ has a good structure, we can start doing some Ergodic Theory. We consider the Teichmuller flow $G_t$ on a stratum $Q_\kappa^{(1)}$. Although Veech showed that the strata are not always connected (see the works of Kontsevich, Zorich and Lanneau for the complete classification of the connected components), Masur and Veech (independently) managed to show that $G_t$ is ergodic on each connected component of $Q_\kappa$ and, more recently, Avila, Gouezel and Yoccoz proved that $G_t$ is exponentially mixing.

Next, we ask about the Lyapounov spectrum (i.e., the collection of Lyapounov exponents) of $G_t$. In order to adress properly this question, Kontsevich and Zorich introduced the so-called Kontsevich-Zorich cocycle $G_t^{KZ}:Q_g^{(1)}\times H^1(M,\mathbb{C})\to Q_g^{(1)}\times H^1(M,\mathbb{C})$ given by the quotient of the trivial cocycle $G_t\times id$ by the mapping class group. It is know that $G_t^{KZ}$ is symplectic so that its Lyapounov spectrum with respect to any ergodic measure $\mu$ is symmetric:

$1=\lambda_1^{\mu}\geq \dots\geq \lambda_g^{\mu}\geq 0\geq -\lambda_g^{\mu}\geq \dots \geq -\lambda_1^{\mu}=-1$.

Moreover, one can show that the Lyapounov exponents of $G_t$ are determined by the first $g$ (non-negative) Lyapounov exponents $1=\lambda_1^{\mu}\geq \dots\geq \lambda_g^{\mu}$ of $G_t^{KZ}$.

After several numerical experiments, Kontsevich and Zorich conjectured that the Lyapounov exponents of $G_t^{KZ}$ with respect to the canonical ‘Lebesgue’ measure $\mu_\kappa^{(1)}$ are all non-zero (i.e., $G_t$ is non-uniformly hyperbolic) and simple (i.e., multiplicity 1). Nowadays, we know that this conjecture is true due to the results of G. Forni (who proved the non-uniform hyperbolicity of $G_t^{KZ}$) and A. Avila, M. Viana (who showed that the simplicity of the Lyapounov spectrum). In particular, the Kontsevich-Zorich spectrum (i.e., the Lyapounov spectrum of $G_t^{KZ}$) of a $\mu_\kappa^{(1)}$ generic point is well-understood.

A natural question (posed by Veech) related to this result concerns the Kontsevich-Zorich spectrum of non-generic points: how bad (or ‘degenerate’) can it be? This question was firstly answered by G. Forni who showed the existence of a Veech surface of genus 3 so that any $SL(2,\mathbb{R})$-invariant measure $\mu$ supported on the $SL(2,\mathbb{R})$-orbit of this surface has $\lambda_2^{\mu}=\lambda_3^{\mu}=0$ (moreover, it seems that there are no such examples in genus 2).

At this point, we are ready to state the following result:

Theorem(G. Forni, —). Any $SL(2,\mathbb{R})$-invariant measure $\mu$ supported on the $SL(2,\mathbb{R})$-orbit of the genus 4 Riemann surface associated to the algebraic equation

$w^6=(z-x_1)(z-x_2)(z-x_3)(z-x_4)^3$

has ‘degenerate’ Kontsevich-Zorich spectrum: $\lambda_2^{\mu}=\lambda_3^{\mu}=\lambda_4^{\mu}=0$.

Roughly speaking, the basic idea here is: this Riemann surface is ‘sufficiently symmetric’ (i.e., it has a ‘good’ automorphism group); on the other hand, by Forni’s method, the presence of symmetries to show some cancellations of the Lyapounov exponents (and the more symmetries you have, the more cancellations you get); finally, in the case of this surface, it turns out that the cyclic group of automorphisms generated by the symmetry $T(z,w)=(z,\varepsilon_6\cdot w)$ (where $\varepsilon_6$ is a 6-th root of unity) suffices to completely annilihate the Lyapounov spectrum (except for the ‘trivial’ exponent $\lambda_1^{\mu}=1$).

Remark 1. It turns out that Forni’s method automatically implies that the cocycle $G_t^{KZ}$ along the $SL(2,\mathbb{R})$-orbit of this surface is isometric (and moreover the cocycle is trivial in the sense that it is conjugated to constant). This is an interesting phenomenon if you compare with the ‘chaotic‘ behavior exhibited by generic points.

Remark 2. It is not hard to see (via simple arithmetic arguments) that the method of our paper does not produce any new examples of the type

$w^N=\prod\limits_{n=1}^{4}(z-x_n)^{a_n}$

where $0, $\sum\limits_{n=1}^4 a_n\equiv 0 (mod N)$ and $gcd(a_1,\dots,a_4,N)=1$ (the first two conditions are imposed to guarantee good symmetries while the third condition is necessary to get a connected Riemann surface). On the other hand, M. Möller told us that there are no further such examples among Veech surfaces (in any genus).

Of course, this theorem is just the tip of the iceberg: for instance, it would be interesting to know whether one can find examples of surfaces with a prescribed number of non-zero Lyapounov exponents (that is, given any $0, there is a surface with $\lambda_r^{\mu}>0=\lambda_{r+1}^{\mu}=\dots=\lambda_g^{\mu}$?). It is worth to observe that Forni’s method has a weak point: while we can detect surfaces with all (but one) Lyapounov exponents equal to zero, we can’t prove that a prescribed part of the spectrum vanishes for a given surface and the basic reason is the lack of an explicit formula for the sum of the first $r exponents although Forni has an explicit formula for the sum of all exponents. Currently, I’m working with my post-doc advisor J.-C. Yoccoz in order to come around this problem by the usage of other methods, but this is still a work in progress… So, I think here is a good point to end this post!

I hope to see you soon in the proof of Asaoka’s theorem! Bye!