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

(We introduce two square-tiled surfaces, one with 8 squares inside ΩM₃(2, 2), and the other with 9 squares inside ΩM₄(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.)