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Dynamics of the Teichmüller geodesic flow on the moduli space of curves and asymptotic monodromy of the Hodge bundle along this flow have numerous applications to dynamics and geometry of measured foliations, to billiards in polygons, to interval exchange transformations, and to geometry of flat surfaces.
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Howard Masur, Martin Möller, Anton Zorich, Flat Surfaces and Dynamics on Moduli Space. Oberwolfach Rep. 11 (2014), no. 1, pp. 869–941DOI 10.4171/OWR/2014/15