# American Institute of Mathematical Sciences

April  2011, 5(2): 355-395. doi: 10.3934/jmd.2011.5.355

## A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle

 1 Department of Mathematics, University of Maryland, College Park, MD 20742-4015

Received  October 2010 Revised  April 2011 Published  July 2011

We establish a geometric criterion on a $SL(2, R)$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $SL(2, R)$-orbits of all algebraically primitive Veech surfaces (see also [7]) and of all Prym eigenforms discovered in [34], as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also [4, 17]). The argument simplifies and generalizes our proof for the case of canonical measures [17]. In the Appendix, Carlos Matheus discusses several relevant examples which further illustrate the power and the limitations of our criterion.
Citation: Giovanni Forni. A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle. Journal of Modern Dynamics, 2011, 5 (2) : 355-395. doi: 10.3934/jmd.2011.5.355
##### References:
 [1] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, "Lattice Point Asymptotics and Volume Growth on Teichmüller Space," (2010), 1-39, arXiv:math/0610715. Google Scholar [2] J. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards, Duke Math. J., 144 (2008), 285-319. doi: 10.1215/00127094-2008-037.  Google Scholar [3] A. Avila and G. Forni, Weak mixing for interval-exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.  Google Scholar [4] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Kontsevich-Zorich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1.  Google Scholar [5] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geometry & Topology, 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.  Google Scholar [6] M. Bainbridge and M. Möller, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus three,, 2009, (): 1.   Google Scholar [7] 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 [8] A. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum, (2009), 1-29, arXiv:0902.3303v1. Google Scholar [9] \bysame, Limit Theorems for Translation Flows, (2010), 1-69, arXiv:0804.3970v3. Google Scholar [10] K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8.  Google Scholar [11] A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 355-395. doi: 10.3934/jmd.2011.5.355.  Google Scholar [12] \bysame, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., ().   Google Scholar [13] A. Eskin and M. Mirzakhani, On invariant and stationary measures for the $\SL(2,R)$ action on moduli space, preprint, 2010. Google Scholar [14] H. M. Farkas and I. Kra, "Riemann Surfaces," Second edition, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992.  Google Scholar [15] J. D. Fay, "Theta Functions on Riemann Surfaces," Lecture Notes in Mathematics, 352, Springer-Verlag, Berlin-New York, 1973.  Google Scholar [16] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464.  Google Scholar [17] \bysame, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  Google Scholar [18] \bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, Handbook of Dynamical Systems Vol. 1B, (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, (2006), 549-580.  Google Scholar [19] G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with degenerate Kontsevich-Zorich spectrum, (2008), 1-8, arXiv:0810.0023. Google Scholar [20] 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 [21] \bysame, Lyapunov spectrum of equivariant subbundles of the Hodge bundle, preprint, 2010. Google Scholar [22] F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.  Google Scholar [23] P. Hubert and E. Lanneau, Veech groups without parabolic elements, Duke Math. J., 133 (2006), 335-346. doi: 10.1215/S0012-7094-06-13326-4.  Google Scholar [24] P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2001), 461-495.  Google Scholar [25] A. B. Katok, Invariant measures of flows on oriented surfaces, Dokl. Nauk. SSR 211, 1973, pp. 775-778 (English translation: Sov. Math. Dokl. 14, 1973, pp. 1104-1108). Google Scholar [26] R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000) 65-108. doi: 10.1007/s000140050113.  Google Scholar [27] M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Scientific Publ., River Edge, NJ, (1997), 318-332.  Google Scholar [28] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x.  Google Scholar [29] R. Krikorian, Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich (d'après Forni, Kontsevich, Zorich, ...), (French) [Deviations of ergodic averages, Teichmüller flows and the Kontsevich-Zorich cocycle (following Forni, Kontsevich, Zorick, ...)], Séminaire Bourbaki, 2003/2004 (2005), 59-93.  Google Scholar [30] H. Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2), 102 (1975), 205-221. doi: 10.2307/1971031.  Google Scholar [31] \bysame, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.  Google Scholar [32] C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486. doi: 10.3934/jmd.2010.4.453.  Google Scholar [33] C. McMullen, Dynamics of $SL_2(\R)$ over moduli space in genus two, Ann. of Math. (2), 165 (2007), 397-456. doi: 10.4007/annals.2007.165.397.  Google Scholar [34] \bysame, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar [35] \bysame, Braid groups and Hodge theory,, preprint, (): 1.   Google Scholar [36] M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., 165 (2006), 633-649.  Google Scholar [37] \bysame, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.  Google Scholar [38] A. Nevo and E. M. Stein, Analogs of Wiener's ergodic theorems for semisimple groups. I, Ann. of Math. (2), 145 (1997), 565-595. doi: 10.2307/2951845.  Google Scholar [39] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar [40] J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260. doi: 10.1007/BF02771535.  Google Scholar [41] R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, preprint, 2010, arXiv:1010.1038v2. Google Scholar [42] W. Veech, Gauss measures for transformations on the space of interval-exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar [43] \bysame, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.  Google Scholar [44] \bysame, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar [45] \bysame, The Forni cocycle, J. Mod. Dyn. 2 (2008), 375-395. doi: 10.3934/jmd.2008.2.375.  Google Scholar [46] A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143. doi: 10.2996/kmj/1138036124.  Google Scholar [47] A. Zorich, Asymptotic flag of an orientable measured foliation on a surface, in "Geometric Study of Foliations" (Tokyo, 1993), World Scientific Publ., River Edge, NJ, (1994), 479-498.  Google Scholar [48] \bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.  Google Scholar [49] \bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.  Google Scholar [50] \bysame, On hyperplane sections of periodic surfaces, in "Solitons, Geometry and Topology: On the Crossroad" (eds. V.M. Buchstaber and S.P. Novikov), Amer. Math. Soc. Transl. (2), 179, AMS, Providence, RI, (1997), 173-189.  Google Scholar [51] \bysame, How do the leaves of a closed $1$-form wind around a surface?, in "Pseudoperiodic Topology" (eds. V.I. Arnol'd, M. Kontsevich and A. Zorich), Amer. Math. Soc. Transl. (2), 197, AMS, Providence, RI, (1999), 135-178.  Google Scholar [52] \bysame, "Flat Surfaces," Frontiers in Number Theory, Physics, and Geometry I, Springer, Berlin, (2006), 437-583.  Google Scholar

show all references

##### References:
 [1] J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, "Lattice Point Asymptotics and Volume Growth on Teichmüller Space," (2010), 1-39, arXiv:math/0610715. Google Scholar [2] J. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards, Duke Math. J., 144 (2008), 285-319. doi: 10.1215/00127094-2008-037.  Google Scholar [3] A. Avila and G. Forni, Weak mixing for interval-exchange transformations and translation flows, Ann. of Math. (2), 165 (2007), 637-664. doi: 10.4007/annals.2007.165.637.  Google Scholar [4] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Kontsevich-Zorich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1.  Google Scholar [5] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geometry & Topology, 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.  Google Scholar [6] M. Bainbridge and M. Möller, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus three,, 2009, (): 1.   Google Scholar [7] 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 [8] A. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum, (2009), 1-29, arXiv:0902.3303v1. Google Scholar [9] \bysame, Limit Theorems for Translation Flows, (2010), 1-69, arXiv:0804.3970v3. Google Scholar [10] K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8.  Google Scholar [11] A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 355-395. doi: 10.3934/jmd.2011.5.355.  Google Scholar [12] \bysame, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., ().   Google Scholar [13] A. Eskin and M. Mirzakhani, On invariant and stationary measures for the $\SL(2,R)$ action on moduli space, preprint, 2010. Google Scholar [14] H. M. Farkas and I. Kra, "Riemann Surfaces," Second edition, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992.  Google Scholar [15] J. D. Fay, "Theta Functions on Riemann Surfaces," Lecture Notes in Mathematics, 352, Springer-Verlag, Berlin-New York, 1973.  Google Scholar [16] G. Forni, Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus, Ann. of Math. (2), 146 (1997), 295-344. doi: 10.2307/2952464.  Google Scholar [17] \bysame, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.  Google Scholar [18] \bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle, Handbook of Dynamical Systems Vol. 1B, (eds. B. Hasselblatt and A. Katok), Elsevier B. V., Amsterdam, (2006), 549-580.  Google Scholar [19] G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with degenerate Kontsevich-Zorich spectrum, (2008), 1-8, arXiv:0810.0023. Google Scholar [20] 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 [21] \bysame, Lyapunov spectrum of equivariant subbundles of the Hodge bundle, preprint, 2010. Google Scholar [22] F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.  Google Scholar [23] P. Hubert and E. Lanneau, Veech groups without parabolic elements, Duke Math. J., 133 (2006), 335-346. doi: 10.1215/S0012-7094-06-13326-4.  Google Scholar [24] P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2001), 461-495.  Google Scholar [25] A. B. Katok, Invariant measures of flows on oriented surfaces, Dokl. Nauk. SSR 211, 1973, pp. 775-778 (English translation: Sov. Math. Dokl. 14, 1973, pp. 1104-1108). Google Scholar [26] R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000) 65-108. doi: 10.1007/s000140050113.  Google Scholar [27] M. Kontsevich, Lyapunov exponents and Hodge theory, in "The Mathematical Beauty of Physics" (Saclay, 1996), Adv. Ser. Math. Phys., 24, World Scientific Publ., River Edge, NJ, (1997), 318-332.  Google Scholar [28] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678. doi: 10.1007/s00222-003-0303-x.  Google Scholar [29] R. Krikorian, Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich (d'après Forni, Kontsevich, Zorich, ...), (French) [Deviations of ergodic averages, Teichmüller flows and the Kontsevich-Zorich cocycle (following Forni, Kontsevich, Zorick, ...)], Séminaire Bourbaki, 2003/2004 (2005), 59-93.  Google Scholar [30] H. Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2), 102 (1975), 205-221. doi: 10.2307/1971031.  Google Scholar [31] \bysame, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.  Google Scholar [32] C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis, J. Mod. Dyn., 4 (2010), 453-486. doi: 10.3934/jmd.2010.4.453.  Google Scholar [33] C. McMullen, Dynamics of $SL_2(\R)$ over moduli space in genus two, Ann. of Math. (2), 165 (2007), 397-456. doi: 10.4007/annals.2007.165.397.  Google Scholar [34] \bysame, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.  Google Scholar [35] \bysame, Braid groups and Hodge theory,, preprint, (): 1.   Google Scholar [36] M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., 165 (2006), 633-649.  Google Scholar [37] \bysame, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.  Google Scholar [38] A. Nevo and E. M. Stein, Analogs of Wiener's ergodic theorems for semisimple groups. I, Ann. of Math. (2), 145 (1997), 565-595. doi: 10.2307/2951845.  Google Scholar [39] G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar [40] J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260. doi: 10.1007/BF02771535.  Google Scholar [41] R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, preprint, 2010, arXiv:1010.1038v2. Google Scholar [42] W. Veech, Gauss measures for transformations on the space of interval-exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar [43] \bysame, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.  Google Scholar [44] \bysame, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.  Google Scholar [45] \bysame, The Forni cocycle, J. Mod. Dyn. 2 (2008), 375-395. doi: 10.3934/jmd.2008.2.375.  Google Scholar [46] A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143. doi: 10.2996/kmj/1138036124.  Google Scholar [47] A. Zorich, Asymptotic flag of an orientable measured foliation on a surface, in "Geometric Study of Foliations" (Tokyo, 1993), World Scientific Publ., River Edge, NJ, (1994), 479-498.  Google Scholar [48] \bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.  Google Scholar [49] \bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.  Google Scholar [50] \bysame, On hyperplane sections of periodic surfaces, in "Solitons, Geometry and Topology: On the Crossroad" (eds. V.M. Buchstaber and S.P. Novikov), Amer. Math. Soc. Transl. (2), 179, AMS, Providence, RI, (1997), 173-189.  Google Scholar [51] \bysame, How do the leaves of a closed $1$-form wind around a surface?, in "Pseudoperiodic Topology" (eds. V.I. Arnol'd, M. Kontsevich and A. Zorich), Amer. Math. Soc. Transl. (2), 197, AMS, Providence, RI, (1999), 135-178.  Google Scholar [52] \bysame, "Flat Surfaces," Frontiers in Number Theory, Physics, and Geometry I, Springer, Berlin, (2006), 437-583.  Google Scholar
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