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The Teichmüller geodesic flow and the geometry of the Hodge bundle. (English) Zbl 1375.37102
Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2010–2011. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 29, 73-95 (2011).
Summary: The Teichmüller geodesic flow is the flow obtained by quasiconformal deformation of Riemann surface structures. The goal of this lecture is to show the strong connection between the geometry of the Hodge bundle (a vector bundle over the moduli space of Riemann surfaces) and the dynamics of the Teichmüller geodesic flow. In particular, we shall provide geometric criterions (based on the variational formulas derived by G. Forni) to detect some special orbits (“totally degenerate”) of the Teichmüller geodesic flow. These results have been obtained jointly with J.-C. Yoccoz [J. Mod. Dyn. 4, No. 3, 453–486 (2010; Zbl 1220.37004)] and G. Forni and A. Zorich [J. Mod. Dyn. 5, No. 2, 285–318 (2011; Zbl 1276.37021); Ergodic Theory Dyn. Syst. 34, No. 2, 353–408 (2014; Zbl 1290.37002)].
For the entire collection see [Zbl 1356.35008].
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53D25 Geodesic flows in symplectic geometry and contact geometry
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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