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Theses

Classification des composantes connexes des strates de l'espace des modules des différentielles quadratiques

Abstract : In this thesis, we study dynamics of the Teichmüller geodesic flow. The origin of this study lies of a very important class of dynamical systems, namely the so-called interval exchange transformations. In classical works in 1982, Masur and Veech has discovered that dynamics of those transformations of the interval is related with the dynamics of the Teichmüller geodesic flow on the moduli spaces of complex curves. The phase space of this flow can be seen as the moduli space of quadratic differentials on a surface. These spaces are naturally stratified by the type of singularities of those differentials. Morever, these strata are preserved by the action of the Teichmüller geodesic flow. Classical results say that these strata carry a natural complex orbifolds structure and are non-empty and non-connected in « general ». The motivation of this work in this thesis come from a fundamental result, proved independently by Masur and by Veech (1982), which says that the Teichmüller geodesic flow acts ergodically on each connected component of each (normalized) stratum, with respect to a finite mass measure preserving. Kontsevich and Zorich have classified the connected components of the strata of the moduli space of Abelian differentials Hg. In this thesis, we give the precis description of the connected components of the strata in the complementary case of Kontsevich-Zorich; that is we describe component of any stratum of the moduli space of quadratic differentials Qg which are not the global square of any Abelian differential. Moreover, we give an explicit formula for the parity of the spin structure of a quadratic differentials inside Qg in terms of the singularities of the stratum. This contradicts a conjecture of Kontsevich-Zorich on the classification of the set the non-hyperelliptic components of Qg by this spin structure. Using this formula, we give an application in the context of rational billiard.
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https://tel.archives-ouvertes.fr/tel-00007204
Contributor : Marie-Annick Guillemer <>
Submitted on : Thursday, October 28, 2004 - 3:08:30 PM
Last modification on : Friday, July 10, 2020 - 4:05:04 PM
Long-term archiving on: : Wednesday, November 23, 2016 - 5:21:35 PM

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  • HAL Id : tel-00007204, version 2

Citation

Erwan Lanneau. Classification des composantes connexes des strates de l'espace des modules des différentielles quadratiques. Mathématiques [math]. Université Rennes 1, 2003. Français. ⟨tel-00007204v2⟩

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