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On the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for quadratic differentials

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Abstract

We prove the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for a measure supported on abelian differentials which come from non-orientable quadratic differentials through a standard orienting, double cover construction. The proof uses Forni’s criterion (J. Mod. Dyn. 5(2):355–395, 2011) for non-uniform hyperbolicity of the cocycle for \({SL(2, \mathbb{R})}\)-invariant measures. We apply these results to the study of deviations in homology of typical leaves of the vertical and horizontal (non-orientable) foliations and deviations of ergodic averages.

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References

  1. Avila, A., Resende, J.M.: Exponential mixing for the Teichmuller flow in the space of quadratic differentials, comment. Math. Helv. (to appear)

  2. Avila A., Viana M.: Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. Acta Math. 198(1), 1–56 (2007) [MRMR2316268 (2008m:37010)]

    Article  MathSciNet  MATH  Google Scholar 

  3. Bainbridge M.: Euler characteristics of Teichmüller curves in genus two. Geom. Topol. 11, 1887–2073 (2007) [MR2350471 (2009c:32025)]

    Article  MathSciNet  MATH  Google Scholar 

  4. Boissy C., Lanneau E.: Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials. Ergodic Theory Dyn. Syst. 29(3), 767–816 (2009) [MRMR2505317 (2010g:37050)]

    Article  MathSciNet  MATH  Google Scholar 

  5. Bufetov, A.I.: Limit theorems for translation flows, ArXiv e-prints (2008)

  6. Delecroix, V., Hubert, P., Lelièvre, S.: Diffusion for the periodic wind-tree model, ArXiv e-prints (2011)

  7. Eckmann J.P., Ruelle D.: Ergodic theory of chaos and strange attractors. Rev. Modern Phys. 57(3, part 1), 617–656 (1985) [MR800052 (87d:58083a)]

    Article  MathSciNet  Google Scholar 

  8. Fickenscher, J.: Self-inverses in rauzy classes, ArXiv e-prints (2011)

  9. Forni G., Matheus C., Zorich A.: Square-tiled cyclic covers. J. Mod. Dyn. 5(2), 285–318 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Forni G.: Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. Math. (2) 155(1), 1–103 (2002) [MR MR1888794 (2003g:37009)]

    Article  MathSciNet  MATH  Google Scholar 

  11. Forni, G.: Sobolev regularity of solutions of the cohomological equation, ArXiv e-prints (2007)

  12. Forni G.: A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle. J. Mod. Dyn. 5(2), 355–395 (2011) [with an appendix by Carlos Matheus. MR2820565]

    Article  MathSciNet  MATH  Google Scholar 

  13. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems, Encyclopedia of mathematics and its applications, vol. 54, Cambridge University Press, Cambridge (1995) (with a supplementary chapter by Katok and Leonardo Mendoza) [MRMR1326374 (96c:58055)]

  14. Kontsevich M.: Lyapunov exponents and Hodge theory, the mathematical beauty of physics (Saclay, 1996). Adv. Ser. Math. Phys. 24, 318–332 (1997) [MRMR1490861 (99b:58147)]

    MathSciNet  Google Scholar 

  15. Kontsevich M., Zorich A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math. 153(3), 631–678 (2003) [MRMR2000471 (2005b:32030)]

    Article  MathSciNet  MATH  Google Scholar 

  16. Lanneau, E.: Parity of the Spin structure defined by a quadratic differential. Geom. Topol. 8, 511–538 (2004) (electronic) [MRMR2057772 (2005h:53149)]

  17. Lanneau E.: Connected components of the strata of the moduli spaces of quadratic differentials. Ann. Sci. Éc. Norm. Supér. (4) 41(1), 1–56 (2008) [MRMR2423309 (2009e:30094)]

    MathSciNet  MATH  Google Scholar 

  18. Masur H.: Interval exchange transformations and measured foliations. Ann. Math. (2) 115(1), 169–200 (1982) [MRMR644018 (83e:28012)]

    Article  MathSciNet  MATH  Google Scholar 

  19. Masur H., Smillie J.: Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, comment. Math. Helv. 68(2), 289–307 (1993) [MRMR1214233 (94d:32028)]

    Article  MathSciNet  MATH  Google Scholar 

  20. Schwartzman S.: Asymptotic cycles. Ann. Math. (2) 66, 270–284 (1957) [MRMR0088720 (19,568i)]

    Article  MathSciNet  MATH  Google Scholar 

  21. Veech W.A.: The Teichmüller geodesic flow. Ann. Math. (2) 124(3), 441–530 (1986) [MR866707 (88g:58153)]

    Article  MathSciNet  MATH  Google Scholar 

  22. Zorich A.: How do the leaves of a closed 1-form wind around a surface? Pseudoperiodic topology. Am. Math. Soc. Trans. Ser. 2 197, 135–178 (1999) [MRMR1733872 (2001c:57019)]

    MathSciNet  Google Scholar 

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  1. Department of Mathematics, The University of Maryland, College Park, MD, 20742, USA

    Rodrigo Treviño

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  1. Rodrigo Treviño
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Correspondence to Rodrigo Treviño.

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Supported by the Brin and Flagship Fellowships at the University of Maryland.

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Treviño, R. On the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for quadratic differentials. Geom Dedicata 163, 311–338 (2013). https://doi.org/10.1007/s10711-012-9751-z

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  • DOI: https://doi.org/10.1007/s10711-012-9751-z

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