Lyapunov Exponents of Rank 2-Variations of Hodge Structures and Modular Embeddings
[Exposants de Lyapunov de variations de structures de Hodge de rang 2 et plongements modulaires]
Kappes, André
Annales de l'Institut Fourier, Tome 64 (2014), p. 2037-2066 / Harvested from Numdam
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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.

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.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/aif.2903
Classification:  32G20,  37D25,  30F35
Mots clés: Exposants de Lyapunov, cocycle de Kontsevich-Zorich, variations de structures de Hodge 
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     author = {Kappes, Andr\'e},
     title = {Lyapunov Exponents of Rank $2$-Variations of Hodge Structures and Modular Embeddings},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {2037-2066},
     doi = {10.5802/aif.2903},
     zbl = {06387330},
     mrnumber = {3330930},
     zbl = {1314.32020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_5_2037_0}
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Kappes, André. Lyapunov Exponents of Rank $2$-Variations of Hodge Structures and Modular Embeddings. Annales de l'Institut Fourier, Tome 64 (2014) pp. 2037-2066. doi : 10.5802/aif.2903. http://gdmltest.u-ga.fr/item/AIF_2014__64_5_2037_0/

[1] Bainbridge, M. Euler characteristics of Teichmüller curves in genus two, Geom. Topol., Tome 11 (2007), pp. 1887-2073 | Article | MR 2350471 | Zbl 1131.32007

[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), Tome 172 (2010) no. 1, pp. 139-185 | Article | MR 2680418 | Zbl 1203.37049

[4] Carlson, J.; Müller-Stach, S.; Peters, C. Period mappings and period domains, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 85 (2003), pp. xvi+430 | MR 2012297 | Zbl 1030.14004

[5] Cohen, P.; Wolfart, J. Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith., Tome 56 (1990) no. 2, pp. 93-110 | MR 1075639 | Zbl 0717.14014

[6] Deligne, P. Un théorèeme de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven, Conn., 1984), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 67 (1987), pp. 1-19 | MR 900821 | Zbl 0656.14010

[7] Ellenberg, Jordan S. Endomorphism algebras of Jacobians, Adv. Math., Tome 162 (2001) no. 2, pp. 243-271 | Article | MR 1859248 | Zbl 1065.14507

[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 3270590 | Zbl 1305.32007

[9] Eskin, Alex; Kontsevich, Maxim; Zorich, Anton Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., Tome 5 (2011) no. 2, pp. 319-353 | Article | MR 2820564 | Zbl 1254.32019

[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), Tome 155 (2002) no. 1, pp. 1-103 | Article | MR 1888794 | Zbl 1034.37003

[12] Gutkin, E.; Judge, C. Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J., Tome 103 (2000) no. 2, pp. 191-213 | Article | MR 1760625 | Zbl 0965.30019

[13] Herrlich, Frank Teichmüller curves defined by characteristic origamis, The geometry of Riemann surfaces and abelian varieties, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 397 (2006), pp. 133-144 | Article | MR 2218004 | Zbl 1098.14019

[14] Herrlich, Frank; Schmithüsen, Gabriela On the boundary of Teichmüller disks in Teichmüller and in Schottky space, Handbook of Teichmüller theory. Vol. I, Eur. Math. Soc., Zürich (IRMA Lect. Math. Theor. Phys.) Tome 11 (2007), pp. 293-349 | Article | MR 2349673 | Zbl 1141.30011

[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 | Article | MR 2186246 | Zbl 1130.37367

[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 1058.37508

[19] Kontsevich, Maxim; Zorich, Anton Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Tome 153 (2003) no. 3, pp. 631-678 | Article | MR 2000471 | Zbl 1087.32010

[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., Tome 16 (2003) no. 4, pp. 857-885 | Article | MR 1992827 | Zbl 1030.32012

[22] Möller, M. Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc., Tome 19 (2006) no. 2, p. 327-344 (electronic) | Article | MR 2188128 | Zbl 1090.32004

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

[24] Schmithüsen, G. An algorithm for finding the Veech group of an origami, Experiment. Math., Tome 13 (2004) no. 4, pp. 459-472 http://projecteuclid.org/getRecord?id=euclid.em/1109106438 | Article | MR 2118271 | Zbl 1078.14036

[25] Shiga, H. On holomorphic mappings of complex manifolds with ball model, J. Math. Soc. Japan, Tome 56 (2004) no. 4, pp. 1087-1107 | Article | MR 2091418 | Zbl 1066.32022

[26] Stillwell, John Classical topology and combinatorial group theory, Springer, New York, Graduate texts in mathematics ; 72 (1980) | MR 602149 | Zbl 0453.57001

[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., Tome 3 (2012) no. 1, pp. 405-426 | Article | MR 2988814 | Zbl 1254.32021

[29] Wright, Alex Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller-Teichmüller curves, Geom. Funct. Anal., Tome 23 (2013) no. 2, pp. 776-809 | Article | MR 3053761 | Zbl 1267.30099

[30] Zimmer, Robert J. Ergodic theory and semisimple groups, Birkhäuser Verlag, Basel, Monographs in Mathematics, Tome 81 (1984), pp. x+209 | MR 776417 | Zbl 0571.58015

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