Un contre-exemple \`a la r\'eciproque du crit\`ere de Forni pour la positivit\'e des exposants de Lyapunov du cocycle de Kontsevich-Zorich

Abstract

We introduce two square-tiled surfaces, one with $8$ squares inside $\Omega \mathcal{M}_3(2,2)$, and the other with $9$ squares inside $\Omega \mathcal{M}_4(3,3)$, respectively. In these examples, the dimensions of the isotropic subspaces (in absolute homology) generated by the waist curves of the maximal cylinders in any fixed rational direction are $2$ and $3$ respectively. Hence, a geometrical criterion of G. Forni for the non-uniform hyperbolicity of Kontsevich-Zorich (KZ) cocycle can not be applied to these examples. Nevertheless, we prove that there are no vanishing exponents and the spectrum is simple for these two square-tiled surfaces. In particular, the non-vanishing of exponents of KZ cocycle for a regular measure doesn't imply that the support of this measure contains a completely periodic surface whose waist curves of maximal cylinders generates a Lagrangian subspace in its absolute homology.