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

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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

Giovanni Forni ^1,

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

Received October 2010 Revised April 2011 Published July 2011

Abstract
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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.

Keywords: Teichmüller geodesic flow, Kontsevich--Zorich cocycle, non-uniform hyperbolicity., moduli space of abelian differentials.

Mathematics Subject Classification: Primary: 30F30, 32G15, 32G20, 57M50; Secondary: 14D07, 37D2.

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.
[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.
[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.
[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.
[5]	M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geometry & Topology, 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.
[6]	M. Bainbridge and M. Möller, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus three,, 2009, (): 1.
[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.
[8]	A. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum, (2009), 1-29, arXiv:0902.3303v1.
[9]	\bysame, Limit Theorems for Translation Flows, (2010), 1-69, arXiv:0804.3970v3.
[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.
[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.
[12]	\bysame, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., ().
[13]	A. Eskin and M. Mirzakhani, On invariant and stationary measures for the $\SL(2,R)$ action on moduli space, preprint, 2010.
[14]	H. M. Farkas and I. Kra, "Riemann Surfaces," Second edition, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992.
[15]	J. D. Fay, "Theta Functions on Riemann Surfaces," Lecture Notes in Mathematics, 352, Springer-Verlag, Berlin-New York, 1973.
[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.
[17]	\bysame, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.
[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.
[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.
[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.
[21]	\bysame, Lyapunov spectrum of equivariant subbundles of the Hodge bundle, preprint, 2010.
[22]	F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.
[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.
[24]	P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2001), 461-495.
[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).
[26]	R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000) 65-108. doi: 10.1007/s000140050113.
[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.
[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.
[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.
[30]	H. Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2), 102 (1975), 205-221. doi: 10.2307/1971031.
[31]	\bysame, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.
[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.
[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.
[34]	\bysame, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.
[35]	\bysame, Braid groups and Hodge theory,, preprint, (): 1.
[36]	M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., 165 (2006), 633-649.
[37]	\bysame, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.
[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.
[39]	G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
[40]	J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260. doi: 10.1007/BF02771535.
[41]	R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, preprint, 2010, arXiv:1010.1038v2.
[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.
[43]	\bysame, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.
[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.
[45]	\bysame, The Forni cocycle, J. Mod. Dyn. 2 (2008), 375-395. doi: 10.3934/jmd.2008.2.375.
[46]	A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143. doi: 10.2996/kmj/1138036124.
[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.
[48]	\bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.
[49]	\bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.
[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.
[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.
[52]	\bysame, "Flat Surfaces," Frontiers in Number Theory, Physics, and Geometry I, Springer, Berlin, (2006), 437-583.

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.
[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.
[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.
[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.
[5]	M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geometry & Topology, 11 (2007), 1887-2073. doi: 10.2140/gt.2007.11.1887.
[6]	M. Bainbridge and M. Möller, The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus three,, 2009, (): 1.
[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.
[8]	A. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum, (2009), 1-29, arXiv:0902.3303v1.
[9]	\bysame, Limit Theorems for Translation Flows, (2010), 1-69, arXiv:0804.3970v3.
[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.
[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.
[12]	\bysame, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., ().
[13]	A. Eskin and M. Mirzakhani, On invariant and stationary measures for the $\SL(2,R)$ action on moduli space, preprint, 2010.
[14]	H. M. Farkas and I. Kra, "Riemann Surfaces," Second edition, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1992.
[15]	J. D. Fay, "Theta Functions on Riemann Surfaces," Lecture Notes in Mathematics, 352, Springer-Verlag, Berlin-New York, 1973.
[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.
[17]	\bysame, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2), 155 (2002), 1-103.
[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.
[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.
[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.
[21]	\bysame, Lyapunov spectrum of equivariant subbundles of the Hodge bundle, preprint, 2010.
[22]	F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr., 281 (2008), 219-237. doi: 10.1002/mana.200510597.
[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.
[24]	P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2001), 461-495.
[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).
[26]	R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv., 75 (2000) 65-108. doi: 10.1007/s000140050113.
[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.
[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.
[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.
[30]	H. Masur, On a class of geodesics in Teichmüller space, Ann. of Math. (2), 102 (1975), 205-221. doi: 10.2307/1971031.
[31]	\bysame, Interval-exchange transformations and measured foliations, Ann. of Math. (2), 115 (1982), 169-200.
[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.
[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.
[34]	\bysame, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590. doi: 10.1215/S0012-7094-06-13335-5.
[35]	\bysame, Braid groups and Hodge theory,, preprint, (): 1.
[36]	M. Möller, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., 165 (2006), 633-649.
[37]	\bysame, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32. doi: 10.3934/jmd.2011.5.1.
[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.
[39]	G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
[40]	J. Smillie and B. Weiss, Minimal sets for flows on moduli space, Israel J. Math., 142 (2004), 249-260. doi: 10.1007/BF02771535.
[41]	R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials, preprint, 2010, arXiv:1010.1038v2.
[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.
[43]	\bysame, The Teichmüller geodesic flow, Ann. of Math. (2), 124 (1986), 441-530.
[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.
[45]	\bysame, The Forni cocycle, J. Mod. Dyn. 2 (2008), 375-395. doi: 10.3934/jmd.2008.2.375.
[46]	A. Yamada, Precise variational formulas for abelian differentials, Kodai Math. J., 3 (1980), 114-143. doi: 10.2996/kmj/1138036124.
[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.
[48]	\bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.
[49]	\bysame, Deviation for interval-exchange transformations, Ergod. Th. & Dynam. Sys., 17 (1997), 1477-1499.
[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.
[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.
[52]	\bysame, "Flat Surfaces," Frontiers in Number Theory, Physics, and Geometry I, Springer, Berlin, (2006), 437-583.

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