Mathematics – Dynamical Systems
Scientific paper
2011-03-08
Mathematics
Dynamical Systems
5 pages, 2 figures. Abstracts in French and English. Main text in French. The arguments were slightly reworked to fit the new
Scientific paper
We study two square-tiled surfaces, one with 8 squares (and genus 3), and other with 9 squares (and genus 4), resp. 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 resp. Hence, a geometrical criterion of G. Forni for the non-uniform hyperbolicity of Kontsevich-Zorich (KZ) cocycle can't 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.
Delecroix Vincent
Matheus Carlos
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