Lyapunov Exponents of Rank 2 -Variations of Hodge Structures and Modular Embeddings

André Kappes[1]

  • [1] Goethe-Universität Frankfurt am Main Institut für Mathematik Robert-Mayer-Str. 6–8 Frankfurt am Main (Germany)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 5, page 2037-2066
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Kappes, André. "Lyapunov Exponents of Rank $2$-Variations of Hodge Structures and Modular Embeddings." Annales de l’institut Fourier 64.5 (2014): 2037-2066. <http://eudml.org/doc/275414>.

@article{Kappes2014,
abstract = {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.},
affiliation = {Goethe-Universität Frankfurt am Main Institut für Mathematik Robert-Mayer-Str. 6–8 Frankfurt am Main (Germany)},
author = {Kappes, André},
journal = {Annales de l’institut Fourier},
keywords = {Lyapunov exponent; Kontsevich-Zorich cocycle; variations of Hodge structures},
language = {eng},
number = {5},
pages = {2037-2066},
publisher = {Association des Annales de l’institut Fourier},
title = {Lyapunov Exponents of Rank $2$-Variations of Hodge Structures and Modular Embeddings},
url = {http://eudml.org/doc/275414},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Kappes, André
TI - Lyapunov Exponents of Rank $2$-Variations of Hodge Structures and Modular Embeddings
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 5
SP - 2037
EP - 2066
AB - 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.
LA - eng
KW - Lyapunov exponent; Kontsevich-Zorich cocycle; variations of Hodge structures
UR - http://eudml.org/doc/275414
ER -

References

top
  1. M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007), 1887-2073 Zbl1131.32007MR2350471
  2. Oliver Bauer, Familien von Jacobivarietäten über Origamikurven, (2009), Universitätsverlag, Karlsruhe 
  3. I. Bouw, M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172 (2010), 139-185 Zbl1203.37049MR2680418
  4. J. Carlson, S. Müller-Stach, C. Peters, Period mappings and period domains, 85 (2003), Cambridge University Press, Cambridge Zbl1030.14004MR2012297
  5. P. Cohen, J. Wolfart, Modular embeddings for some nonarithmetic Fuchsian groups, Acta Arith. 56 (1990), 93-110 Zbl0717.14014MR1075639
  6. P. Deligne, Un théorèeme de finitude pour la monodromie, Discrete groups in geometry and analysis (New Haven, Conn., 1984) 67 (1987), 1-19, Birkhäuser Boston, Boston, MA Zbl0656.14010MR900821
  7. Jordan S. Ellenberg, Endomorphism algebras of Jacobians, Adv. Math. 162 (2001), 243-271 Zbl1065.14507MR1859248
  8. A. Eskin, M. Kontsevich, A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, (2010) Zbl1305.32007
  9. Alex Eskin, Maxim Kontsevich, Anton Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn. 5 (2011), 319-353 Zbl1254.32019MR2820564
  10. Myriam Finster, Stabilisatorgruppen in Aut ( F z ) und Veechgruppen von Überlagerungen, (2008) 
  11. G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), 1-103 Zbl1034.37003MR1888794
  12. E. Gutkin, C. Judge, Affine mappings of translation surfaces: geometry and arithmetic, Duke Math. J. 103 (2000), 191-213 Zbl0965.30019MR1760625
  13. Frank Herrlich, Teichmüller curves defined by characteristic origamis, The geometry of Riemann surfaces and abelian varieties 397 (2006), 133-144, Amer. Math. Soc., Providence, RI Zbl1098.14019MR2218004
  14. Frank Herrlich, Gabriela Schmithüsen, On the boundary of Teichmüller disks in Teichmüller and in Schottky space, Handbook of Teichmüller theory. Vol. I 11 (2007), 293-349, Eur. Math. Soc., Zürich Zbl1141.30011MR2349673
  15. Pascal Hubert, Thomas A. Schmidt, An introduction to Veech surfaces, Handbook of dynamical systems. Vol. 1B (2006), 501-526, Elsevier B. V., Amsterdam Zbl1130.37367MR2186246
  16. André Kappes, Monodromy Representations and Lyapunov Exponents of Origamis, (2011) 
  17. André Kappes, Martin Möller, Lyapunov spectrum of ball quotients with applications to commensurability questions, (2012) Zbl1334.22010
  18. M. Kontsevich, A. Zorich, Lyapunov exponents and Hodge theory, (1997) Zbl1058.37508
  19. Maxim Kontsevich, Anton Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631-678 Zbl1087.32010MR2000471
  20. C. Matheus, J.-C. Yoccoz, D. Zmiaikou, Homology of origamis with symmetries, (2012) Zbl06387303
  21. C. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), 857-885 Zbl1030.32012MR1992827
  22. M. Möller, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), 327-344 (electronic) Zbl1090.32004MR2188128
  23. M. Möller, Teichmüller curves, mainly from the point of view of algebraic geometry, Moduli spaces of Riemann surfaces 20 (2013), 267-318, Amer. Math. Soc., Providence, RI Zbl1279.14031
  24. G. Schmithüsen, An algorithm for finding the Veech group of an origami, Experiment. Math. 13 (2004), 459-472 Zbl1078.14036MR2118271
  25. H. Shiga, On holomorphic mappings of complex manifolds with ball model, J. Math. Soc. Japan 56 (2004), 1087-1107 Zbl1066.32022MR2091418
  26. John Stillwell, Classical topology and combinatorial group theory, (1980), Springer, New York Zbl0453.57001MR602149
  27. C. Weiss, Twisted Teichmüller curves, (2012) 
  28. Alex Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn. 3 (2012), 405-426 Zbl1254.32021MR2988814
  29. Alex Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller-Teichmüller curves, Geom. Funct. Anal. 23 (2013), 776-809 Zbl1267.30099MR3053761
  30. Robert J. Zimmer, Ergodic theory and semisimple groups, 81 (1984), Birkhäuser Verlag, Basel Zbl0571.58015MR776417
  31. David Zmiaikou, Origamis and permutation groups, (2011) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.