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), (2010), 1.

[2]

J. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards,, Duke Math. J., 144 (2008), 285. 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. 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. doi: 10.1007/s11511-007-0012-1.

[5]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two,, Geometry & Topology, 11 (2007), 1887. 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. doi: 10.4007/annals.2010.172.139.

[8]

A. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum,, (2009), (2009), 1.

[9]

\bysame, Limit Theorems for Translation Flows,, (2010), (2010), 1.

[10]

K. Calta, Veech surfaces and complete periodicity in genus two,, J. Amer. Math. Soc., 17 (2004), 871. 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. 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, 71 (1992).

[15]

J. D. Fay, "Theta Functions on Riemann Surfaces,", Lecture Notes in Mathematics, 352 (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. 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.

[18]

\bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, Handbook of Dynamical Systems Vol. 1B, (2006), 549.

[19]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with degenerate Kontsevich-Zorich spectrum,, (2008), (2008), 1.

[20]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. 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. doi: 10.1002/mana.200510597.

[23]

P. Hubert and E. Lanneau, Veech groups without parabolic elements,, Duke Math. J., 133 (2006), 335. 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.

[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, 14 (1973), 1104.

[26]

R. Kenyon and J. Smillie, Billiards on rational-angled triangles,, Comment. Math. Helv., 75 (2000), 65. doi: 10.1007/s000140050113.

[27]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318.

[28]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. 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, 2003/2004 (2005), 59.

[30]

H. Masur, On a class of geodesics in Teichmüller space,, Ann. of Math. (2), 102 (1975), 205. doi: 10.2307/1971031.

[31]

\bysame, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169.

[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. 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. doi: 10.4007/annals.2007.165.397.

[34]

\bysame, Prym varieties and Teichmüller curves,, Duke Math. J., 133 (2006), 569. 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.

[37]

\bysame, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. 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. doi: 10.2307/2951845.

[39]

G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315.

[40]

J. Smillie and B. Weiss, Minimal sets for flows on moduli space,, Israel J. Math., 142 (2004), 249. doi: 10.1007/BF02771535.

[41]

R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials,, preprint, (2010).

[42]

W. Veech, Gauss measures for transformations on the space of interval-exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391.

[43]

\bysame, The Teichmüller geodesic flow,, Ann. of Math. (2), 124 (1986), 441.

[44]

\bysame, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 553. doi: 10.1007/BF01388890.

[45]

\bysame, The Forni cocycle,, J. Mod. Dyn. 2 (2008), 2 (2008), 375. doi: 10.3934/jmd.2008.2.375.

[46]

A. Yamada, Precise variational formulas for abelian differentials,, Kodai Math. J., 3 (1980), 114. doi: 10.2996/kmj/1138036124.

[47]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1994), 479.

[48]

\bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents,, Ann. Inst. Fourier (Grenoble), 46 (1996), 325.

[49]

\bysame, Deviation for interval-exchange transformations,, Ergod. Th. & Dynam. Sys., 17 (1997), 1477.

[50]

\bysame, On hyperplane sections of periodic surfaces,, in, 179 (1997), 173.

[51]

\bysame, How do the leaves of a closed $1$-form wind around a surface?,, in, 197 (1999), 135.

[52]

\bysame, "Flat Surfaces,", Frontiers in Number Theory, (2006), 437.

show all references

References:
[1]

J. Athreya, A. Bufetov, A. Eskin and M. Mirzakhani, "Lattice Point Asymptotics and Volume Growth on Teichmüller Space,", (2010), (2010), 1.

[2]

J. Athreya and G. Forni, Deviation of ergodic averages for rational polygonal billiards,, Duke Math. J., 144 (2008), 285. 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. 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. doi: 10.1007/s11511-007-0012-1.

[5]

M. Bainbridge, Euler characteristics of Teichmüller curves in genus two,, Geometry & Topology, 11 (2007), 1887. 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. doi: 10.4007/annals.2010.172.139.

[8]

A. Bufetov, Finitely-additive measures on the asymptotic foliations of a Markov compactum,, (2009), (2009), 1.

[9]

\bysame, Limit Theorems for Translation Flows,, (2010), (2010), 1.

[10]

K. Calta, Veech surfaces and complete periodicity in genus two,, J. Amer. Math. Soc., 17 (2004), 871. 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. 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, 71 (1992).

[15]

J. D. Fay, "Theta Functions on Riemann Surfaces,", Lecture Notes in Mathematics, 352 (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. 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.

[18]

\bysame, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, Handbook of Dynamical Systems Vol. 1B, (2006), 549.

[19]

G. Forni and C. Matheus, An example of a Teichmüller disk in genus $4$ with degenerate Kontsevich-Zorich spectrum,, (2008), (2008), 1.

[20]

G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. 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. doi: 10.1002/mana.200510597.

[23]

P. Hubert and E. Lanneau, Veech groups without parabolic elements,, Duke Math. J., 133 (2006), 335. 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.

[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, 14 (1973), 1104.

[26]

R. Kenyon and J. Smillie, Billiards on rational-angled triangles,, Comment. Math. Helv., 75 (2000), 65. doi: 10.1007/s000140050113.

[27]

M. Kontsevich, Lyapunov exponents and Hodge theory,, in, 24 (1997), 318.

[28]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,, Invent. Math., 153 (2003), 631. 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, 2003/2004 (2005), 59.

[30]

H. Masur, On a class of geodesics in Teichmüller space,, Ann. of Math. (2), 102 (1975), 205. doi: 10.2307/1971031.

[31]

\bysame, Interval-exchange transformations and measured foliations,, Ann. of Math. (2), 115 (1982), 169.

[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. 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. doi: 10.4007/annals.2007.165.397.

[34]

\bysame, Prym varieties and Teichmüller curves,, Duke Math. J., 133 (2006), 569. 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.

[37]

\bysame, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. 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. doi: 10.2307/2951845.

[39]

G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315.

[40]

J. Smillie and B. Weiss, Minimal sets for flows on moduli space,, Israel J. Math., 142 (2004), 249. doi: 10.1007/BF02771535.

[41]

R. Treviño, On the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for quadratic differentials,, preprint, (2010).

[42]

W. Veech, Gauss measures for transformations on the space of interval-exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391.

[43]

\bysame, The Teichmüller geodesic flow,, Ann. of Math. (2), 124 (1986), 441.

[44]

\bysame, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards,, Invent. Math., 97 (1989), 553. doi: 10.1007/BF01388890.

[45]

\bysame, The Forni cocycle,, J. Mod. Dyn. 2 (2008), 2 (2008), 375. doi: 10.3934/jmd.2008.2.375.

[46]

A. Yamada, Precise variational formulas for abelian differentials,, Kodai Math. J., 3 (1980), 114. doi: 10.2996/kmj/1138036124.

[47]

A. Zorich, Asymptotic flag of an orientable measured foliation on a surface,, in, (1994), 479.

[48]

\bysame, Finite Gauss measure on the space of interval-exchange transformations. Lyapunov exponents,, Ann. Inst. Fourier (Grenoble), 46 (1996), 325.

[49]

\bysame, Deviation for interval-exchange transformations,, Ergod. Th. & Dynam. Sys., 17 (1997), 1477.

[50]

\bysame, On hyperplane sections of periodic surfaces,, in, 179 (1997), 173.

[51]

\bysame, How do the leaves of a closed $1$-form wind around a surface?,, in, 197 (1999), 135.

[52]

\bysame, "Flat Surfaces,", Frontiers in Number Theory, (2006), 437.

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