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Lyapunov spectrum of invariant subbundles of the Hodge bundle

  • GIOVANNI FORNI (a1), CARLOS MATHEUS (a2) and ANTON ZORICH (a3)
Abstract

We study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (the Kodaira–Spencer map) of the Hodge bundle with respect to the Gauss–Manin connection and investigate the relations between the central Oseledets subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.

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[AEZ]Athreya, J., Eskin, A. and Zorich, A.. Right-angled billiards and volumes of the moduli spaces of quadratic differentials on ℂP1, in preparation.
[Au]Aulicino, D.. Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum. Preprint, 2012, 1–74, arXiv:1205.2359.
[AV]Avila, A. and Viana, M.. Simplicity of Lyapunov spectra: proof of the Kontsevich–Zorich conjecture. Acta Math. 198 (2007), 1–56.
[Ba]Bainbridge, M.. Euler characteristics of Teichmüller curves in genus two. Geom. Topol. 11 (2007), 1887–2073.
[BDV]Bonatti, C., Diaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102). Springer, New York, 2005.
[B]Bouw, I.. The p-rank of ramified covers of curves. Compositio Math. 126 (2001), 295–322.
[BMo]Bouw, I. and Möller, M.. Teichmüller curves, triangle groups and Lyapunov exponents. Ann. of Math. (2) 172 (2010), 139–185.
[DHL]Delecroix, V., Hubert, P. and Lelièvre, S.. Diffusion for the periodic wind-tree model. Preprint, 2011, pp. 1–28, arXiv:1107.1810.
[EKZ1]Eskin, A., Kontsevich, M. and Zorich, A.. Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Preprint, 2011, pp. 1–106, arXiv:1112.5872.
[EKZ2]Eskin, A., Kontsevich, M. and Zorich, A.. Lyapunov spectrum of square-tiled cyclic covers. J. Mod. Dyn. 5(2) (2011), 319–353.
[F1]Forni, G.. Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. of Math. (2) 146(2) (1997), 295–344.
[F2]Forni, G.. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2) 155(1) (2002), 1–103.
[F3]Forni, G.. On the Lyapunov exponents of the Kontsevich–Zorich cocycle (Handbook of Dynamical Systems, 1B). Eds. Hasselblatt, B. and Katok, A.. Elsevier, 2006, pp. 549–580.
[F4]Forni, G.. A geometric criterion for the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle (with an appendix by C. Matheus). J. Mod. Dyn. 5(2) (2011), 355–395.
[FMt]Forni, G. and Matheus, C.. An example of a Teichmüller disk in genus 4 with degenerate Kontsevich–Zorich spectrum. Preprint, 2008, pp. 1–8, arXiv:0810.0023.
[FMZ]Forni, G., Matheus, C. and Zorich, A.. Square-tiled cyclic covers. J. Mod. Dyn. 5(2) (2011), 285–318.
[FMZ2]Forni, G., Matheus, C. and Zorich, A.. Zero Lyapunov exponents of the Hodge bundle. Comment. Math. Helv. (2012), 1–39, to appear, arXiv:1201.6075.
[GH]Griffiths, Ph. and Harris, J.. Principles of Algebraic Geometry. Wiley, New York, 1978.
[HK]Hasselblatt, B. and Katok, A.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and Its Applications, 54). Cambridge University Press, Cambridge, 1995.
[HS]Herrlich, F. and Schmithüsen, G.. An extraordinary origami curve. Math. Nachr. 281(2) (2008), 219–237.
[IT]Imayoshi, Y. and Taniguchi, M.. An Introduction to Teichmüller Spaces. Springer, Tokyo, 1992.
[KM]Kappes, A. and Möller, M.. Lyapunov spectrum of ball quotients with applications to commensurability questions. Preprint, 2012, pp. 1–37, arXiv:1207.5433.
[K]Kontsevich, M.. Lyapunov exponents and Hodge theory. The Mathematical Beauty of Physics, Saclay, 1996 (Advanced Series in Mathematical Physics, 24). World Scientific, River Edge, NJ, 1997, pp. 318–332.
[M]Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.
[MT]Masur, H. and Tabachnikov, S.. Rational Billiards and Flat Structures (Handbook of Dynamical Systems, 1A). North-Holland, Amsterdam, 2002, pp. 1015–1089.
[Mc]McMullen, C.. Braid groups and Hodge theory, Math. Ann. to appear. Preprint, 2009, http://www.springerlink.com/content/h564377036829147.
[Mo]Möller, M.. Shimura and Teichmüller curves. J. Mod. Dyn. 5(1) (2011), 1–32.
[Na]Nag, S.. The Complex Analytic Theory of Teichmüller Spaces. John Wiley, New York, 1988.
[Ro]Rokhlin, V.. On the fundamental ideas of measure theory. Mat. Sb. N.S. 25(67) (1949), 107–150.
[V]Veech, W.. The Teichmüller geodesic flow. Ann. of Math. 124 (1986), 441–530.
[Z0]Zorich, A.. Asymptotic flag of an orientable measured foliation on a surface. Geometric Study of Foliations (Tokyo, 1993). World Scientific, River Edge, NJ, 1994, pp. 479–498.
[Z1]Zorich, A.. How do the leaves of a closed 1-form wind around a surface. In the collection: ‘Pseudoperiodic Topology’ (American Mathematical Society Translations, Series 2, 197). American Mathematical Society, Providence, RI, 1999, pp. 135–178.
[Z2]Zorich, A.. Flat surfaces. In the collection ‘Frontiers in Number Theory, Physics and Geometry. Vol. 1: On Random Matrices, Zeta Functions and Dynamical systems’; École de Physique des Houches, France, March 9–21 2003. Eds. Cartier, P., Julia, B., Moussa, P. and Vanhove, P.. Springer, Berlin, 2006, pp. 439–586.
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Ergodic Theory and Dynamical Systems
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  • EISSN: 1469-4417
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