Weierstrass filtration on Teichmüller curves and Lyapunov exponents

Previous Article
On cyclicity-one elliptic islands of the standard map
JMD Home
This Issue
Next Article
Infinitely many lattice surfaces with special pseudo-Anosov maps

April 2013, 7(2): 209-237. doi: 10.3934/jmd.2013.7.209

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

Received September 2012 Published September 2013

Abstract
Related Papers

We define the Weierstrass filtration for Teichmüller curves and construct the Harder-Narasimhan filtration of the Hodge bundle of a Teichmüller curve in hyperelliptic loci and low-genus nonvarying strata. As a result we obtain the sum of Lyapunov exponents of Teichmüller curves in these strata.

Keywords: Teichmüller curves, Harder-Narasimhan filtration, Lyapunov exponents.

Mathematics Subject Classification: Primary: 32G15; Secondary: 14H1.

Citation: Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209

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 (1985). Google Scholar
[2]	M. Bainbridge, Euler characteristics of Teichmüller curves in genus two,, Geom. Topol., 11 (2007), 1887. doi: 10.2140/gt.2007.11.1887. Google Scholar
[3]	E. M. Bullock, "Subcanonical Points on Algebraic Curves,'', Ph.D. Thesis, (2009). Google Scholar
[4]	I. Bouw and M. Möller, Teichmüller cuves, triangle groups, and Lyapunov exponents,, Ann of Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139. Google Scholar
[5]	D. Chen, Square-tiled surfaces and rigid curves on moduli spaces,, Adv. Math., 228 (2011), 1135. doi: 10.1016/j.aim.2011.06.002. Google Scholar
[6]	D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus,, Geom. Topol., 16 (2012), 2427. Google Scholar
[7]	D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., (). Google Scholar
[8]	A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319. Google Scholar
[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}., (). Google Scholar
[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. doi: 10.1007/s10240-003-0015-1. Google Scholar
[11]	G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar
[12]	R. Hartshorne, "Algebraic Geometry,'', Graduate Texts in Mathematics, (1977). Google Scholar
[13]	D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'', Aspects of Mathematics, E31 (1997). Google Scholar
[14]	M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., (). Google Scholar
[15]	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. Google Scholar
[16]	E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., (). Google Scholar
[17]	C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces,, J. Amer. Math. Soc., 16 (2003), 857. doi: 10.1090/S0894-0347-03-00432-6. Google Scholar
[18]	C. T. McMullen, Prym varieties and Teichmüller curves,, Duke Math. J., 133 (2006), 569. doi: 10.1215/S0012-7094-06-13335-5. Google Scholar
[19]	C. T. McMullen, Foliations of Hilbert modular surfaces,, Amer. J. Math., 129 (2007), 183. doi: 10.1353/ajm.2007.0002. Google Scholar
[20]	M. Möller, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. doi: 10.3934/jmd.2011.5.1. Google Scholar
[21]	M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., (). Google Scholar
[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}., (). Google Scholar
[23]	E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties,, J. Diff. Geometry, 66 (2004), 233. Google Scholar
[24]	G. Xiao, Fibered algebraic surfaces with low slope,, Math. Ann., 276 (1987), 449. doi: 10.1007/BF01450841. Google Scholar
[25]	F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., (). Google Scholar

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 (1985). Google Scholar
[2]	M. Bainbridge, Euler characteristics of Teichmüller curves in genus two,, Geom. Topol., 11 (2007), 1887. doi: 10.2140/gt.2007.11.1887. Google Scholar
[3]	E. M. Bullock, "Subcanonical Points on Algebraic Curves,'', Ph.D. Thesis, (2009). Google Scholar
[4]	I. Bouw and M. Möller, Teichmüller cuves, triangle groups, and Lyapunov exponents,, Ann of Math. (2), 172 (2010), 139. doi: 10.4007/annals.2010.172.139. Google Scholar
[5]	D. Chen, Square-tiled surfaces and rigid curves on moduli spaces,, Adv. Math., 228 (2011), 1135. doi: 10.1016/j.aim.2011.06.002. Google Scholar
[6]	D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus,, Geom. Topol., 16 (2012), 2427. Google Scholar
[7]	D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, to appear in Ann. Sci. École Norm. Sup., (). Google Scholar
[8]	A. Eskin, M. Kontsevich and A. Zorich, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319. Google Scholar
[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}., (). Google Scholar
[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. doi: 10.1007/s10240-003-0015-1. Google Scholar
[11]	G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 285. doi: 10.3934/jmd.2011.5.285. Google Scholar
[12]	R. Hartshorne, "Algebraic Geometry,'', Graduate Texts in Mathematics, (1977). Google Scholar
[13]	D. Huybrechts and M. Lehn, "The Geometry of Moduli Spaces of Sheaves,'', Aspects of Mathematics, E31 (1997). Google Scholar
[14]	M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory,, \arXiv{hep-th/9701164}., (). Google Scholar
[15]	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. Google Scholar
[16]	E. Lanneau and D.-N. Manh, Teichmüller curves generated by Weierstraß Prym eigenforms in genus three,, \arXiv{1111.2299}., (). Google Scholar
[17]	C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces,, J. Amer. Math. Soc., 16 (2003), 857. doi: 10.1090/S0894-0347-03-00432-6. Google Scholar
[18]	C. T. McMullen, Prym varieties and Teichmüller curves,, Duke Math. J., 133 (2006), 569. doi: 10.1215/S0012-7094-06-13335-5. Google Scholar
[19]	C. T. McMullen, Foliations of Hilbert modular surfaces,, Amer. J. Math., 129 (2007), 183. doi: 10.1353/ajm.2007.0002. Google Scholar
[20]	M. Möller, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. doi: 10.3934/jmd.2011.5.1. Google Scholar
[21]	M. Möller, Prym covers, theta functions and Kobayashi curves in Hilbert modular surfaces,, to appear in Amer. Journal of Math., (). Google Scholar
[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}., (). Google Scholar
[23]	E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties,, J. Diff. Geometry, 66 (2004), 233. Google Scholar
[24]	G. Xiao, Fibered algebraic surfaces with low slope,, Math. Ann., 276 (1987), 449. doi: 10.1007/BF01450841. Google Scholar
[25]	F. Yu and K. Zuo, Weierstrass filtration on Teichmüller curves and Lyapunov exponents: Upper bound,, \arXiv{1209.2733}., (). Google Scholar

[1]	Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1
[2]	Alex Wright. Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces. Journal of Modern Dynamics, 2012, 6 (3) : 405-426. doi: 10.3934/jmd.2012.6.405
[3]	Dawei Chen. Strata of abelian differentials and the Teichmüller dynamics. Journal of Modern Dynamics, 2013, 7 (1) : 135-152. doi: 10.3934/jmd.2013.7.135
[4]	Ursula Hamenstädt. Bowen's construction for the Teichmüller flow. Journal of Modern Dynamics, 2013, 7 (4) : 489-526. doi: 10.3934/jmd.2013.7.489
[5]	Ursula Hamenstädt. Dynamics of the Teichmüller flow on compact invariant sets. Journal of Modern Dynamics, 2010, 4 (2) : 393-418. doi: 10.3934/jmd.2010.4.393
[6]	Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535
[7]	Chi Po Choi, Xianfeng Gu, Lok Ming Lui. Subdivision connectivity remeshing via Teichmüller extremal map. Inverse Problems & Imaging, 2017, 11 (5) : 825-855. doi: 10.3934/ipi.2017039
[8]	Matteo Costantini, André Kappes. The equation of the Kenyon-Smillie (2, 3, 4)-Teichmüller curve. Journal of Modern Dynamics, 2017, 11: 17-41. doi: 10.3934/jmd.2017002
[9]	Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[10]	Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[11]	Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[12]	Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[13]	Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
[14]	Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[15]	Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
[16]	Giovanni Forni. On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations. Journal of Modern Dynamics, 2012, 6 (2) : 139-182. doi: 10.3934/jmd.2012.6.139
[17]	Anna Lenzhen, Babak Modami, Kasra Rafi. Teichmüller geodesics with $ d$-dimensional limit sets. Journal of Modern Dynamics, 2018, 12: 261-283. doi: 10.3934/jmd.2018010
[18]	Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[19]	Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[20]	Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149

2018 Impact Factor: 0.295

American Institute of Mathematical Sciences