Abstract
We prove the Zorich–Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller ow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.
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Avila, A., Forni, G.: Weak mixing for interval exchange transformations and translation flows. Ann. of Math., 165 (2007)
Avila, A., Viana, M.: Dynamics in the moduli space of abelian differentials. Port. Math., 62, 531–547 (2005)
Avila, A., Viana, M.: Simplicity of Lyapunov spectra: a sufficient criterion. To appear in Port. Math
Bonatti, C., Gómez-Mont, X., Viana, M.: Généricité d’exposants de Lyapunov non-nuls pour des produits déterministes de matrices. Ann. Inst. H. Poincaré Anal. Non Linéaire, 20, 579–624 (2003)
Bonatti, C., Viana, M.: Lyapunov exponents with multiplicity 1 for deterministic products of matrices. Ergodic Theory Dynam. Systems, 24, 1295–1330 (2004)
Forni, G.: Deviation of ergodic averages for area-preserving ows on surfaces of higher genus. Ann. of Math., 155, 1–103 (2002)
Furstenberg, H.: Noncommuting random products. Trans. Amer. Math. Soc., 108, 377–428 (1963)
Goldsheid, I. Y., Margulis, G. A.: Lyapunov exponents of a product of random matrices. Uspekhi Mat. Nauk, 44(5), 13–60 (1989) [(Russian); English translation in Russian Math. Surveys, 44:5 (1989), 11–71]
Guivarc’h, Y., Raugi, A.: Products of random matrices: convergence theorems. In: Random Matrices and their Applications (Brunswick, ME, 1984), Contemp. Math., 50, pp. 31–54. Amer. Math. Soc., Providence, RI, (1986)
Kontsevich, M.: Lyapunov exponents and Hodge theory. In: The Mathematical Beauty of Physics (Saclay, 1996), Adv. Ser. Math. Phys., 24, pp. 318–332. World Scientific, River Edge, NJ (1997)
Kontsevich, M., Zorich, A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math., 153, 631–678 (2003)
Ledrappier, F.: Positivity of the exponent for stationary sequences of matrices. In: Lyapunov Exponents (Bremen, 1984), Lecture Notes in Math., 1186, pp. 56–73. Springer, Berlin, (1986)
Marmi, S., Moussa, P., Yoccoz, J.-C.: The cohomological equation for Roth-type interval exchange maps. J. Amer. Math. Soc., 18, 823–872 (2005)
Masur, H.: Interval exchange transformations and measured foliations. Ann. of Math., 115, 169–200 (1982)
Oseledets, V. I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trudy Moskov. Mat. Obshch., 19, 179–210 (1968) [(Russian); English translation in Trans. Moscow Math. Soc., 19 (1968), 197–231]
Rauzy, G.: Échanges d’intervalles et transformations induites. Acta Arith., 34, 315–328 (1979)
Schwartzman, S.: Asymptotic cycles. Ann. of Math., 66, 270–284 (1957)
Veech, W. A.: Gauss measures for transformations on the space of interval exchange maps. Ann. of Math., 115, 201–242 (1982)
Veech, W. A.: The Teichmüller geodesic ow. Ann. of Math., 124, 441–530 (1986)
Viana, M.: Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents. To appear in Ann. of Math
Zorich, A.: Asymptotic flag of an orientable measured foliation on a surface. In: Geometric Study of Foliations (Tokyo, 1993), pp. 479–498. World Scientific, River Edge, NJ (1994)
Zorich, A.: Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents. Ann. Inst. Fourier (Grenoble), 46, 325–370 (1996)
Zorich, A.: Deviation for interval exchange transformations. Ergodic Theory Dynam. Systems, 17, 1477–1499 (1997)
Zorich, A.: How do the leaves of a closed 1-form wind around a surface?. In: Pseudoperiodic Topology, Amer. Math. Soc. Transl., 197, pp. 135–178. Amer. Math. Soc., Provi-dence, RI, (1999)
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Work carried out within the Brazil–France Agreement in Mathematics. Avila is a Clay Research Fellow. Viana is partially supported by Pronex and Faperj.
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Avila, A., Viana, M. Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture. Acta Math 198, 1–56 (2007). https://doi.org/10.1007/s11511-007-0012-1
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DOI: https://doi.org/10.1007/s11511-007-0012-1