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Lyapunov spectrum of invariant subbundles of the Hodge bundle

Published online by Cambridge University Press:  05 December 2012

GIOVANNI FORNI ,
CARLOS MATHEUS  and
ANTON ZORICH
GIOVANNI FORNI
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA (email: [email protected])
CARLOS MATHEUS
Affiliation:
Université Paris 13, Sorbonne Paris Cité, LAGA CNRS (UMR 7539), F-93430, Villetaneuse, France (email: [email protected], [email protected])
ANTON ZORICH
Affiliation:
Institut de Mathématiques de Jessieu and Institut Universitaire de France, Université Paris 7, France (email: [email protected])
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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.

Type
Review Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 353 - 408
Copyright
©2012 Cambridge University Press 

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References

[AEZ]Athreya, J., Eskin, A. and Zorich, A.. Right-angled billiards and volumes of the moduli spaces of quadratic differentials on ℂP1, in preparation.Google Scholar
[Au]Aulicino, D.. Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum. Preprint, 2012, 1–74, arXiv:1205.2359.Google Scholar
[AV]Avila, A. and Viana, M.. Simplicity of Lyapunov spectra: proof of the Kontsevich–Zorich conjecture. Acta Math. 198 (2007), 1–56.Google Scholar
[Ba]Bainbridge, M.. Euler characteristics of Teichmüller curves in genus two. Geom. Topol. 11 (2007), 1887–2073.Google Scholar
[BDV]Bonatti, C., Diaz, L. and Viana, M.. Dynamics Beyond Uniform Hyperbolicity (Encyclopaedia of Mathematical Sciences, 102). Springer, New York, 2005.Google Scholar
[B]Bouw, I.. The p-rank of ramified covers of curves. Compositio Math. 126 (2001), 295–322.Google Scholar
[BMo]Bouw, I. and Möller, M.. Teichmüller curves, triangle groups and Lyapunov exponents. Ann. of Math. (2) 172 (2010), 139–185.Google Scholar
[DHL]Delecroix, V., Hubert, P. and Lelièvre, S.. Diffusion for the periodic wind-tree model. Preprint, 2011, pp. 1–28, arXiv:1107.1810.Google Scholar
[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.Google Scholar
[EKZ2]Eskin, A., Kontsevich, M. and Zorich, A.. Lyapunov spectrum of square-tiled cyclic covers. J. Mod. Dyn. 5(2) (2011), 319–353.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar
[FMZ]Forni, G., Matheus, C. and Zorich, A.. Square-tiled cyclic covers. J. Mod. Dyn. 5(2) (2011), 285–318.Google Scholar
[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.Google Scholar
[GH]Griffiths, Ph. and Harris, J.. Principles of Algebraic Geometry. Wiley, New York, 1978.Google Scholar
[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.Google Scholar
[HS]Herrlich, F. and Schmithüsen, G.. An extraordinary origami curve. Math. Nachr. 281(2) (2008), 219–237.Google Scholar
[IT]Imayoshi, Y. and Taniguchi, M.. An Introduction to Teichmüller Spaces. Springer, Tokyo, 1992.Google Scholar
[KM]Kappes, A. and Möller, M.. Lyapunov spectrum of ball quotients with applications to commensurability questions. Preprint, 2012, pp. 1–37, arXiv:1207.5433.Google Scholar
[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.Google Scholar
[M]Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.CrossRefGoogle Scholar
[MT]Masur, H. and Tabachnikov, S.. Rational Billiards and Flat Structures (Handbook of Dynamical Systems, 1A). North-Holland, Amsterdam, 2002, pp. 1015–1089.Google Scholar
[Mc]McMullen, C.. Braid groups and Hodge theory, Math. Ann. to appear. Preprint, 2009, http://www.springerlink.com/content/h564377036829147.Google Scholar
[Mo]Möller, M.. Shimura and Teichmüller curves. J. Mod. Dyn. 5(1) (2011), 1–32.Google Scholar
[Na]Nag, S.. The Complex Analytic Theory of Teichmüller Spaces. John Wiley, New York, 1988.Google Scholar
[Ro]Rokhlin, V.. On the fundamental ideas of measure theory. Mat. Sb. N.S. 25(67) (1949), 107–150.Google Scholar
[V]Veech, W.. The Teichmüller geodesic flow. Ann. of Math. 124 (1986), 441–530.Google Scholar
[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.Google Scholar
[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.Google Scholar
[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.Google Scholar