Posted by: matheuscmss | March 9, 2011

## Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich

Today Vincent Delecroix and I uploaded to ArXiv our short note Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich. In this 5 pages article (mainly written in French), we exhibit 2 examples of square-tiled surfaces (of genus 3 and 4 resp.) showing that the reciprocal to G. Forni’s recent non-uniform hyperbolicity criterion of the Kontsevich-Zorich cocycle is not true.

More precisely, let $\mu$ be a $SL(2,\mathbb{R})$-invariant ergodic probability supported on some stratum of the moduli space of Abelian differentials of genus $g$. G. Forni’s criterion says that the Kontsevich-Zorich (KZ) cocycle has no zero exponent with respect to $\mu$ whenever its homological dimension is maximal (i.e., equal to the genus $g$). Here, the homological dimension of $\mu$ is the maximal dimension of isotropic subspaces of absolute homology generated by maximal cylinders of completely periodic Abelian differentials in the support of $\mu$. See these posts for some preliminaries on Abelian differentials, $SL(2,\mathbb{R})$-action and KZ cocycle, and G. Forni’s article for more details on the non-uniform hyperbolicity criterion.

In the short note, we exhibit 2 square-tiled surfaces (of genus 3 and 4 resp.) whose $SL(2,\mathbb{R})$-orbits support ergodic probabilities with homological dimension $g-1$ (where $g$ is the genus), but, nevertheless, with no zero exponent. Actually, we show that there are no repeated Lyapunov exponent (i.e., their multiplicities are 1), that is, the KZ spectrum is simple.

These examples were found through an exhaustive computer search using SAGE: in fact, it was somewhat “shocking” for us to find such a small number (only 2) of square-tiled surfaces with homological dimension strictly less that its genus among a list of several hundreds of square-tiled surfaces in genus 3 and 4. As a matter of fact, since we don’t dispose of a general procedure to construct more examples with low homological dimension, we currently don’t know whether there is an infinite list of them.

Concerning our proof of simplicity of the KZ spectrum of these examples, we simply compute some the homological action of some pseudo-Anosovs in the support of our measures to verify a simplicity criterion for square-tiled surfaces in a work (still in preparation) of M. Möller, J.-C. Yoccoz and myself. In a few words, this simplicity criterion is inspired on previous works of A. Avila and M. Viana on the KZ spectrum of the so-called Masur-Veech measure is studied, even though the monoid of symplectic matrices considered by Avila and Viana (coming from the so-called Rauzy-Veech-Zorich induction) is different from our monoid of symplectic matrices (coming from some sort of covering of the continuous fraction algorithm on the modular surface). Actually, this “change of monoid of matrices” reflects fundamental geometrical differences between the Masur-Veech measure and natural measures supported on $SL(2,\mathbb{R})$-orbits of square-tiled surfaces. I plan to come back to these issues in a future post when some version of the joint work with Möller and Yoccoz is made available publicly.

As a final comment, let me say that, despite our confidence on the simplicity criterion for square-tiled surfaces as stated in our note with Vincent, since we’re still verifying its details, it may happen that Vincent and I will be forced to adapt the calculations in our note to fit an eventually different version of the simplicity criterion.

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