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Lyapunov spectrum of square-tiled cyclic covers
A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle
1. | Department of Mathematics, University of Maryland, College Park, MD 20742-4015 |
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] | |
[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] | |
[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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