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On cyclicity-one elliptic islands of the standard map
Weierstrass filtration on Teichmüller curves and Lyapunov exponents
1. | School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China |
2. | Fachbereich 08-Physik Mathematik und Informatik, Universität Mainz, 55099 Mainz, Germany |
References:
[1] |
E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, "Geometry of Algebraic Curves. Vol. I,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267, Springer-Verlag, New York, 1985. |
[2] |
M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073.
doi: 10.2140/gt.2007.11.1887. |
[3] |
E. M. Bullock, "Subcanonical Points on Algebraic Curves,'' Ph.D. Thesis, Harvard University, 2009. |
[4] |
I. Bouw and M. Möller, Teichmüller cuves, triangle groups, and Lyapunov exponents, Ann of Math. (2), 172 (2010), 139-185.
doi: 10.4007/annals.2010.172.139. |
[5] |
D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228 (2011), 1135-1162.
doi: 10.1016/j.aim.2011.06.002. |
[6] |
D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479. |
[7] |
D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., ().
|
[8] |
A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353.
doi: 10.3934/jmd.2011.5.319. |
[9] |
A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, \arXiv{1112.5872}., ().
|
[10] |
A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems and the Siegel-Veech constats, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179.
doi: 10.1007/s10240-003-0015-1. |
[11] |
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. |
[12] |
R. Hartshorne, "Algebraic Geometry,'' Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. |
[13] |
D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'' Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. |
[14] |
M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., ().
|
[15] |
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. |
[16] |
E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., ().
|
[17] |
C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885.
doi: 10.1090/S0894-0347-03-00432-6. |
[18] |
C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590.
doi: 10.1215/S0012-7094-06-13335-5. |
[19] |
C. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math., 129 (2007), 183-215.
doi: 10.1353/ajm.2007.0002. |
[20] |
M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.
doi: 10.3934/jmd.2011.5.1. |
[21] |
M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., ().
|
[22] |
M. Möller, Teichmüller curves, mainly from the view point of algebraic geometry,, to appear as PCMI Lecture Notes. Available from: \url{http://www.uni-frankfurt.de/fb/fb12/mathematik/ag/personen/moeller/summaries/PCMI.pdf}., ().
|
[23] |
E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties, J. Diff. Geometry, 66 (2004), 233-287. |
[24] |
G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann., 276 (1987), 449-466.
doi: 10.1007/BF01450841. |
[25] |
F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., ().
|
show all references
References:
[1] |
E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, "Geometry of Algebraic Curves. Vol. I,'' Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267, Springer-Verlag, New York, 1985. |
[2] |
M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol., 11 (2007), 1887-2073.
doi: 10.2140/gt.2007.11.1887. |
[3] |
E. M. Bullock, "Subcanonical Points on Algebraic Curves,'' Ph.D. Thesis, Harvard University, 2009. |
[4] |
I. Bouw and M. Möller, Teichmüller cuves, triangle groups, and Lyapunov exponents, Ann of Math. (2), 172 (2010), 139-185.
doi: 10.4007/annals.2010.172.139. |
[5] |
D. Chen, Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228 (2011), 1135-1162.
doi: 10.1016/j.aim.2011.06.002. |
[6] |
D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus, Geom. Topol., 16 (2012), 2427-2479. |
[7] |
D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., ().
|
[8] |
A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5 (2011), 319-353.
doi: 10.3934/jmd.2011.5.319. |
[9] |
A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, \arXiv{1112.5872}., ().
|
[10] |
A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems and the Siegel-Veech constats, Publ. Math. Inst. Hautes Études Sci., 97 (2003), 61-179.
doi: 10.1007/s10240-003-0015-1. |
[11] |
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. |
[12] |
R. Hartshorne, "Algebraic Geometry,'' Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. |
[13] |
D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'' Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. |
[14] |
M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., ().
|
[15] |
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. |
[16] |
E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., ().
|
[17] |
C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc., 16 (2003), 857-885.
doi: 10.1090/S0894-0347-03-00432-6. |
[18] |
C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J., 133 (2006), 569-590.
doi: 10.1215/S0012-7094-06-13335-5. |
[19] |
C. T. McMullen, Foliations of Hilbert modular surfaces, Amer. J. Math., 129 (2007), 183-215.
doi: 10.1353/ajm.2007.0002. |
[20] |
M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn., 5 (2011), 1-32.
doi: 10.3934/jmd.2011.5.1. |
[21] |
M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., ().
|
[22] |
M. Möller, Teichmüller curves, mainly from the view point of algebraic geometry,, to appear as PCMI Lecture Notes. Available from: \url{http://www.uni-frankfurt.de/fb/fb12/mathematik/ag/personen/moeller/summaries/PCMI.pdf}., ().
|
[23] |
E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties, J. Diff. Geometry, 66 (2004), 233-287. |
[24] |
G. Xiao, Fibered algebraic surfaces with low slope, Math. Ann., 276 (1987), 449-466.
doi: 10.1007/BF01450841. |
[25] |
F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., ().
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