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

TL;DR

This paper presents two specific square-tiled surfaces where the usual geometric criterion for non-uniform hyperbolicity of the Kontsevich-Zorich cocycle does not apply, yet the spectrum remains simple and non-vanishing.

Contribution

It provides explicit counterexamples to a geometric criterion for hyperbolicity of the KZ cocycle, showing that non-vanishing exponents can occur without the criterion's conditions.

Findings

No vanishing exponents in the examples.

Spectrum of the KZ cocycle is simple.

Geometric criterion does not imply hyperbolicity in these cases.

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.