Geodesics on Flat Surfaces

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.Comment: (25 pages, 5 figures) Based on the talk at ICM 2006 at Madrid; see Proceedings of the ICM, Madrid, Spain, 2006, EMS, 121-146 for the final version. For a more detailed survey see the paper "Flat Surfaces", arXiv.math.DS/060939

Volumes of strata of Abelian differentials and Siegel-Veech constants in large genera

We state conjectures on the asymptotic behavior of the volumes of moduli spaces of Abelian differentials and their Siegel-Veech constants as genus tends to infinity. We provide certain numerical evidence, describe recent advances and the state of the art towards proving these conjectures.Comment: Some background material is added on request of the referee. To appear in Arnold Math. Journa

Zero Lyapunov exponents of the Hodge bundle

By the results of G. Forni and of R. Trevi\~no, the Lyapunov spectrum of the Hodge bundle over the Teichm\"uller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichm\"uller curve, for which the real Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2,R)-invariant probability measure.Comment: 47 pages, 10 figures. Final version (based on the referee's report). A slightly shorter version of this article will appear in Commentarii Mathematici Helvetici. A pdf file containing a copy of the Mathematica routine "FMZ3-Zariski-numerics_det1.nb" is available at this link here: http://w3.impa.br/~cmateus/files/FMZ3-Zariski-numerics_det1.pd

Lyapunov spectrum of square-tiled cyclic covers

A cyclic cover over the Riemann sphere branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmuller curve. The key technical element is evaluation of degrees of line subbundles of the Hodge bundle, corresponding to eigenspaces of the induced action of deck transformations.Comment: The presentation is simplified. The algebro-geometric background is described more clearly and in more details. Some typos are correcte