Lyapunov Exponents of Rank 2-Variations of Hodge Structures and Modular Embeddings
Annales de l'Institut Fourier, Volume 64 (2014) no. 5, pp. 2037-2066.

If the monodromy representation of a VHS over a hyperbolic curve stabilizes a rank two subspace, there is a single non-negative Lyapunov exponent associated with it. We derive an explicit formula using only the representation in the case when the monodromy is discrete.

Si la représentation de monodromie d’une variation de structures de Hodge sur une courbe hyperbolique stabilise un sous-espace de rang 2, elle possède un seul exposant de Lyapunov non-negative. Nous deduisons une formule explicite pour cet exposant dans le cas où la monodromie est discrète en employant seulement la représentation.

DOI: 10.5802/aif.2903
Classification: 32G20, 37D25, 30F35
Keywords: Lyapunov exponent, Kontsevich-Zorich cocycle, variations of Hodge structures
Mot clés : Exposants de Lyapunov, cocycle de Kontsevich-Zorich, variations de structures de Hodge
Kappes, André 1

1 Goethe-Universität Frankfurt am Main Institut für Mathematik Robert-Mayer-Str. 6–8 Frankfurt am Main (Germany)
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Kappes, André. Lyapunov Exponents of Rank $2$-Variations of Hodge Structures and Modular Embeddings. Annales de l'Institut Fourier, Volume 64 (2014) no. 5, pp. 2037-2066. doi : 10.5802/aif.2903. https://aif.centre-mersenne.org/articles/10.5802/aif.2903/

[1] Bainbridge, M. Euler characteristics of Teichmüller curves in genus two, Geom. Topol., Volume 11 (2007), pp. 1887-2073 | DOI | MR | Zbl

[2] Bauer, Oliver Familien von Jacobivarietäten über Origamikurven, Universitätsverlag, Karlsruhe, 2009

[3] Bouw, I.; Möller, M. Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), Volume 172 (2010) no. 1, pp. 139-185 | DOI | MR | Zbl

[4] Carlson, J.; Müller-Stach, S.; Peters, C. Period mappings and period domains, Cambridge Studies in Advanced Mathematics, 85, Cambridge University Press, Cambridge, 2003, pp. xvi+430 | MR | Zbl

[5] Cohen, P.; Wolfart, J. Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith., Volume 56 (1990) no. 2, pp. 93-110 | EuDML | MR | Zbl

[6] Deligne, P. Un théorèeme de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven, Conn., 1984) (Progr. Math.), Volume 67, Birkhäuser Boston, Boston, MA, 1987, pp. 1-19 | MR | Zbl

[7] Ellenberg, Jordan S. Endomorphism algebras of Jacobians, Adv. Math., Volume 162 (2001) no. 2, pp. 243-271 | DOI | MR | Zbl

[8] Eskin, A.; Kontsevich, M.; Zorich, A. Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow (2010) to appear in Publications de l’IHES (2014) vol. 120, issue 1, arXiv: math.AG/1112.5872 | MR | Zbl

[9] Eskin, Alex; Kontsevich, Maxim; Zorich, Anton Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., Volume 5 (2011) no. 2, pp. 319-353 | DOI | MR | Zbl

[10] Finster, Myriam Stabilisatorgruppen in Aut(F z ) und Veechgruppen von Überlagerungen, Universität Karlsruhe, Fakultät für Mathematik (2008) (diploma thesis)

[11] Forni, G. Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), Volume 155 (2002) no. 1, pp. 1-103 | DOI | MR | Zbl

[12] Gutkin, E.; Judge, C. Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Volume 103 (2000) no. 2, pp. 191-213 | DOI | MR | Zbl

[13] Herrlich, Frank Teichmüller curves defined by characteristic origamis, The geometry of Riemann surfaces and abelian varieties (Contemp. Math.), Volume 397, Amer. Math. Soc., Providence, RI, 2006, pp. 133-144 | DOI | MR | Zbl

[14] Herrlich, Frank; Schmithüsen, Gabriela On the boundary of Teichmüller disks in Teichmüller and in Schottky space, Handbook of Teichmüller theory. Vol. I (IRMA Lect. Math. Theor. Phys.), Volume 11, Eur. Math. Soc., Zürich, 2007, pp. 293-349 | DOI | MR | Zbl

[15] Hubert, Pascal; Schmidt, Thomas A. An introduction to Veech surfaces, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 501-526 | DOI | MR | Zbl

[16] Kappes, André Monodromy Representations and Lyapunov Exponents of Origamis, Karlsruhe Institute of Technology (2011) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000024435 (Ph. D. Thesis)

[17] Kappes, André; Möller, Martin Lyapunov spectrum of ball quotients with applications to commensurability questions, 2012 (to appear in Duke Math. J., arXiv:math/1207.5433)

[18] Kontsevich, M.; Zorich, A. Lyapunov exponents and Hodge theory, 1997 (arXiv:hep-th/9701164) | Zbl

[19] Kontsevich, Maxim; Zorich, Anton Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Volume 153 (2003) no. 3, pp. 631-678 | DOI | MR | Zbl

[20] Matheus, C.; Yoccoz, J.-C.; Zmiaikou, D. Homology of origamis with symmetries (2012) (to appear in Annales de l’Institut Fourier, Volume 64, 2014, arXiv:1207.2423)

[21] McMullen, C. Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., Volume 16 (2003) no. 4, pp. 857-885 | DOI | MR | Zbl

[22] Möller, M. Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc., Volume 19 (2006) no. 2, p. 327-344 (electronic) | DOI | MR | Zbl

[23] Möller, M. Teichmüller curves, mainly from the point of view of algebraic geometry, Moduli spaces of Riemann surfaces (IAS/Park City Math. Ser.), Volume 20, Amer. Math. Soc., Providence, RI, 2013, pp. 267-318

[24] Schmithüsen, G. An algorithm for finding the Veech group of an origami, Experiment. Math., Volume 13 (2004) no. 4, pp. 459-472 http://projecteuclid.org/getRecord?id=euclid.em/1109106438 | DOI | MR | Zbl

[25] Shiga, H. On holomorphic mappings of complex manifolds with ball model, J. Math. Soc. Japan, Volume 56 (2004) no. 4, pp. 1087-1107 | DOI | MR | Zbl

[26] Stillwell, John Classical topology and combinatorial group theory, Graduate texts in mathematics ; 72, Springer, New York, 1980 | MR | Zbl

[27] Weiss, C. Twisted Teichmüller curves, 2012 (Ph.D. Thesis, J. W. Goethe-Universität Frankfurt)

[28] Wright, Alex Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn., Volume 3 (2012) no. 1, pp. 405-426 | DOI | MR | Zbl

[29] Wright, Alex Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller-Teichmüller curves, Geom. Funct. Anal., Volume 23 (2013) no. 2, pp. 776-809 | DOI | MR | Zbl

[30] Zimmer, Robert J. Ergodic theory and semisimple groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984, pp. x+209 | MR | Zbl

[31] Zmiaikou, David Origamis and permutation groups, University Paris-Sud 11, Orsay (2011) (Ph. D. Thesis)

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