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January  2014, 8(1): 61-73. doi: 10.3934/jmd.2014.8.61

Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum

Julien Grivaux 1, and  Pascal Hubert 1,

1. 

I2M, Université d’Aix-Marseille, 39 rue F. Joliot- Curie, 13453 Marseille Cedex 20, France, France

Received  July 2013 Published  July 2014

We construct explicit closed $\mathrm{GL}(2; \mathbb{R})$-invariant loci in strata of meromorphic quadratic differentials of arbitrarily large dimension with fully degenerate Lyapunov spectrum. This answers a question of Forni-Matheus-Zorich.
Keywords: Meromorphic quadratic differentials, degenerate Lyapunov spectrum..
Mathematics Subject Classification: Primary: 30F60, 32G15, 32G20; Secondary: 37H1.
Citation: Julien Grivaux, Pascal Hubert. Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum. Journal of Modern Dynamics, 2014, 8 (1) : 61-73. doi: 10.3934/jmd.2014.8.61
References:
[1]

D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, preprint, arXiv:1302.0913, 2013. Google Scholar

[2]

I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.  Google Scholar

[3]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Osceledets generic in almost every direction, preprint, arXiv:1305.1104, 2013. Google Scholar

[4]

A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319.  Google Scholar

[5]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publications mathématiques de l'IHÉS, (2013), 1-127. doi: 10.1007/s10240-013-0060-3.  Google Scholar

[6]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL(2,R) action on moduli space, preprint, arXiv:1302.3320, 2013. Google Scholar

[7]

H. M. Farkas and I. Kra, Riemann Surfaces, Second edition, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-2034-3.  Google Scholar

[8]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318. doi: 10.3934/jmd.2011.5.285.  Google Scholar

[9]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150.  Google Scholar

[10]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 549-580. doi: 10.1016/S1874-575X(06)80033-0.  Google Scholar

[11]

I. Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math., 146 (1981), 231-270. doi: 10.1007/BF02392465.  Google Scholar

[12]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.  Google Scholar

[13]

A. Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn., 6 (2012), 405-426. doi: 10.3934/jmd.2012.6.405.  Google Scholar

[14]

A. Zorich, Deviation for interval-exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215.  Google Scholar

[15]

A. Zorich, How do the leaves of a closed $1$-form wind around a surface?, in Pseudoperiodic Topology, Amer. Math. Soc. Transl. Ser. 2, 197, Amer. Math. Soc., Providence, RI, 1999, 135-178.  Google Scholar

show all references

References:
[1]

D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, preprint, arXiv:1302.0913, 2013. Google Scholar

[2]

I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.  Google Scholar

[3]

J. Chaika and A. Eskin, Every flat surface is Birkhoff and Osceledets generic in almost every direction, preprint, arXiv:1305.1104, 2013. Google Scholar

[4]

A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353. doi: 10.3934/jmd.2011.5.319.  Google Scholar

[5]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publications mathématiques de l'IHÉS, (2013), 1-127. doi: 10.1007/s10240-013-0060-3.  Google Scholar

[6]

A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL(2,R) action on moduli space, preprint, arXiv:1302.3320, 2013. Google Scholar

[7]

H. M. Farkas and I. Kra, Riemann Surfaces, Second edition, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4612-2034-3.  Google Scholar

[8]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 285-318. doi: 10.3934/jmd.2011.5.285.  Google Scholar

[9]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103. doi: 10.2307/3062150.  Google Scholar

[10]

G. Forni, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, in Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, 2006, 549-580. doi: 10.1016/S1874-575X(06)80033-0.  Google Scholar

[11]

I. Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math., 146 (1981), 231-270. doi: 10.1007/BF02392465.  Google Scholar

[12]

M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.  Google Scholar

[13]

A. Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn., 6 (2012), 405-426. doi: 10.3934/jmd.2012.6.405.  Google Scholar

[14]

A. Zorich, Deviation for interval-exchange transformations, Ergodic Theory Dynam. Systems, 17 (1997), 1477-1499. doi: 10.1017/S0143385797086215.  Google Scholar

[15]

A. Zorich, How do the leaves of a closed $1$-form wind around a surface?, in Pseudoperiodic Topology, Amer. Math. Soc. Transl. Ser. 2, 197, Amer. Math. Soc., Providence, RI, 1999, 135-178.  Google Scholar

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