Duke Mathematical Journal

Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle

Simion Filip

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We describe all the situations in which the Kontsevich–Zorich (KZ) cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, we find this only occurs when the cocycle satisfies additional geometric constraints. We also show that the connected components of the Zariski closure of the monodromy must be from a specific list, and the representations in which they can occur are described. The number of zero exponents of the KZ cocycle is then as small as possible, given its monodromy.

Article information

Duke Math. J., Volume 166, Number 4 (2017), 657-706.

Received: 11 May 2015
Revised: 25 May 2016
First available in Project Euclid: 31 October 2016

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Zentralblatt MATH identifier

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 14D07: Variation of Hodge structures [See also 32G20]

Lyapunov exponents Kontsevich–Zorich cocycle flat surfaces translation surfaces monodromy variations of Hodge structure


Filip, Simion. Zero Lyapunov exponents and monodromy of the Kontsevich–Zorich cocycle. Duke Math. J. 166 (2017), no. 4, 657--706. doi:10.1215/00127094-3715806. https://projecteuclid.org/euclid.dmj/1477918664

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  • [1] Y. André, Mumford-Tate groups of mixed Hodge structures and the theorem of the fixed part, Compos. Math. 82 (1992), 1–24.
  • [2] D. Aulicino, Teichmüller discs with completely degenerate Kontsevich-Zorich spectrum, Comment. Math. Helv. 90 (2015), 573–643.
  • [3] D. Aulicino, Affine invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum, preprint, arXiv:1302.0913v2 [math.DS].
  • [4] A. Avila, A. Eskin, and M. Moeller, Symplectic and Isometric $\mathrm{SL}(2,R)$-invariant subbundles of the Hodge bundle, preprint, arXiv:1209.2854v2 [math.DS].
  • [5] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math. 198 (2007), 1–56.
  • [6] A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math. 181 (2010), 115–189.
  • [7] N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4–6, Elem. Math. (Berlin), Springer, Berlin, 2002.
  • [8] I. I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2) 172 (2010), 139–185.
  • [9] D. Bump, Lie Groups, 2nd ed., Grad. Texts in Math. 225, Springer, New York, 2013.
  • [10] J. Chaika and A. Eskin, Every flat surface is Birkhoff and Oseledets generic in almost every direction, J. Mod. Dyn. 9 (2015), 1–23.
  • [11] P. Deligne, “Travaux de Griffiths” in Séminaire Bourbaki, Vol. 1969/70, Exposé 376, Lecture Notes in Math. 180, Springer, Berlin, 1971. Zbl0208.48601.
  • [12] P. Deligne, La conjecture de Weil pour les surfaces $K3$, Invent. Math. 15 (1972), 206–226.
  • [13] P. Deligne, “Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques” in Automorphic Forms, Representations and $L$-functions (Corvallis, 1977), Part 2, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, 1979, 247–289.
  • [14] P. Deligne, J. S. Milne, A. Ogus, and K. Shih, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900, Springer, Berlin, 1982.
  • [15] A. Eskin, M. Kontsevich, and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn. 5 (2011), 319–353.
  • [16] A. Eskin, M. Kontsevich, and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 207–333.
  • [17] A. Eskin and C. Matheus, A coding-free simplicity criterion for the Lyapunov exponents of Teichmüller curves, Geom. Dedicata 179 (2015), 45–67.
  • [18] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\mathrm{SL}(2,R)$ action on moduli space, preprint, arXiv:1302.3320v4 [math.DS].
  • [19] A. Eskin, M. Mirzakhani, and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $\mathrm{SL}(2,R)$ action on moduli space, Ann. of Math. (2) 182 (2015), 673–721.
  • [20] S. Filip, Semisimplicity and rigidity of the Kontsevich-Zorich cocycle, Invent. Math. 205 (2016), 617–670.
  • [21] S. Filip, Splitting mixed Hodge structures over affine invariant manifolds, Ann. of Math. (2) 183 (2016), 681–713.
  • [22] S. Filip, Families of $K3$ surfaces and Lyapunov exponents, preprint, arXiv:1412.1779v2 [math.DS].
  • [23] S. Filip, G. Forni, and C. Matheus, Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups, preprint, arXiv:1502.07202v2 [math.DS].
  • [24] G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), 1–103.
  • [25] G. Forni, “On the Lyapunov exponents of the Kontsevich-Zorich cocycle” in Handbook of Dynamical Systems, Vol. 1B, Elsevier, Amsterdam, 2006, 549–580.
  • [26] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich-Zorich cocycle, with an appendix by C. Matheus, J. Mod. Dyn. 5 (2011), 355–395.
  • [27] G. Forni and C. Matheus, Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards, J. Mod. Dyn. 8 (2014), 271–436.
  • [28] G. Forni, C. Matheus, and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn. 5 (2011), 285–318.
  • [29] G. Forni, C. Matheus, and A. Zorich, Lyapunov spectrum of invariant subbundles of the Hodge bundle, Ergodic Theory Dynam. Systems 34 (2014), 353–408.
  • [30] G. Forni, C. Matheus, and A. Zorich, Zero Lyapunov exponents of the Hodge bundle, Comment. Math. Helv. 89 (2014), 489–535.
  • [31] I. Y. Gol’dsheĭd and G. A. Margulis, Lyapunov exponents of a product of random matrices (in Russian), Uspekhi Mat. Nauk 44 (1989), no. 5, 13–60; English translation in Russian Math. Surveys 44 (1989), no. 5, 11–71.
  • [32] M. Green, P. Griffiths, and M. Kerr, Mumford-Tate Groups and Domains, Ann. of Math. Stud. 183, Princeton Univ. Press, Princeton, 2012.
  • [33] Y. Guivarc’h and A. Raugi, Propriétés de contraction d’un semi-groupe de matrices inversibles: Coefficients de Liapunoff d’un produit de matrices aléatoires indépendantes, Israel J. Math. 65 (1989), 165–196.
  • [34] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math. 34, Amer. Math. Soc., Providence, 2001.
  • [35] P. Hubert, S. Lelièvre, and S. Troubetzkoy, The Ehrenfest wind-tree model: Periodic directions, recurrence, diffusion, J. Reine Angew. Math. 656 (2011), 223–244.
  • [36] V. A. Kaĭmanovich, Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups (in Russian), Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 164 (1987), Differentsialnaya Geom. Gruppy Li i Mekh. IX, 29–46, 196–197; English translation in J. Soviet Math. 47 (1989), no. 2, 2387–2398.
  • [37] M. Kontsevich, “Lyapunov exponents and Hodge theory” in The Mathematical Beauty of Physics (Saclay, 1996), Adv. Ser. Math. Phys. 24, World Sci. Publ., River Edge, N.J., 1997, 318–332.
  • [38] F. Ledrappier, “Quelques propriétés des exposants caractéristiques” in École d’été de probabilités de Saint-Flour, XII (Saint-Flour, 1982), Lecture Notes in Math. 1097, Springer, Berlin, 1984, 305–396.
  • [39] F. Ledrappier, “Positivity of the exponent for stationary sequences of matrices” in Lyapunov Exponents (Bremen, 1984), Lecture Notes in Math. 1186, Springer, Berlin, 1986, 56–73.
  • [40] J. D. Lewis, A Survey of the Hodge Conjecture, 2nd ed., with an appendix by B. B. Gordon, CRM Monogr. Ser. 10, Amer. Math. Soc., Providence, 1999.
  • [41] E. Looijenga, Moduli spaces and locally symmetric varieties, preprint, arXiv:1404.3854v1 [math.AG].
  • [42] H. Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), 169–200.
  • [43] H. Masur and S. Tabachnikov, “Rational billiards and flat structures” in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015–1089.
  • [44] C. Matheus, J.-C. Yoccoz, and D. Zmiaikou, Homology of origamis with symmetries, Ann. Inst. Fourier (Grenoble) 64 (2014), 1131–1176.
  • [45] C. T. McMullen, Braid groups and Hodge theory, Math. Ann. 355 (2013), 893–946.
  • [46] M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn. 5 (2011), 1–32.
  • [47] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge Structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin, 2008.
  • [48] I. Satake, Holomorphic imbeddings of symmetric domains into a Siegel space, Amer. J. Math. 87 (1965), 425–461.
  • [49] I. Satake, Symplectic representations of algebraic groups satisfying a certain analyticity condition, Acta Math. 117 (1967), 215–279.
  • [50] J.-P. Serre, “Groupes algébriques associés aux modules de Hodge-Tate” in Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. III, Astérisque 65, Soc. Math. France, Paris, 1979, 155–188.
  • [51] J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math. 142 (2004), 249–260.
  • [52] T. A. Springer, Linear Algebraic Groups, Mod. Birkhäuser Class., Birkhäuser, Boston, 2009.
  • [53] R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, Geom. Dedicata 163 (2013), 311–338.
  • [54] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), 201–242.
  • [55] E. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, “Structure of Lie groups and Lie algebras” in Current Problems in Mathematics: Fundamental Directions, Vol. 41, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, 5–259.
  • [56] A. Wright, Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces, J. Mod. Dyn. 6 (2012), 405–426.
  • [57] A. Wright, The field of definition of affine invariant submanifolds of the moduli space of abelian differentials, Geom. Topol. 18 (2014), 1323–1341.
  • [58] A. Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015), 413–438.
  • [59] A. Zorich, Finite Gauss measure on the space of interval exchange transformations: Lyapunov exponents, Ann. Inst. Fourier (Grenoble) 46 (1996), 325–370.
  • [60] 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, 1999, 135–178.
  • [61] A. Zorich, “Flat surfaces” in Frontiers in Number Theory, Physics, and Geometry, I (Les Houches, 2003), Springer, Berlin, 2006, 437–583.