2012, 6(3): 405-426. doi: 10.3934/jmd.2012.6.405

Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces

1. 

Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637, United States

Received  May 2012 Published  October 2012

We consider normal covers of $\mathbb{C}P^1$ with abelian deck group and branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces.
    We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov exponents in terms of the period mapping. In the case of abelian square-tiled surfaces, the Lyapunov exponents are ratios of areas of hyperbolic triangles.
Citation: 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
References:
[1]

F. Beukers, Gauss' hypergeometric functions,, , ().

[2]

______, Notes on differential equations and hypergeometric functions,, , ().

[3]

I. 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.

[4]

J. Carlson, S. Müller-Stach and C. Peters, "Period Mappings and Period Domains,", Cambridge Studies in Advanced Mathematics, 85 (2003).

[5]

T. A. Driscoll and L. N. Trefethen, "Schwarz-Christoffel Mapping,", Cambridge Monographs on Applied and Computational Mathematics, 8 (2002).

[6]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., ().

[7]

______, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319.

[8]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150.

[9]

______, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, (2006), 549.

[10]

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

[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.

[12]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3.

[13]

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

[14]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7.

[15]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis,, Journal of Modern Dynamics, 4 (2010), 453. doi: 10.3934/jmd.2010.4.453.

[16]

M. Möller, Teichmüller curves, Galois actions and $\hat{GT}$-relations,, Math. Nachr., 278 (2005), 1061. doi: 10.1002/mana.200310292.

[17]

______, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. doi: 10.3934/jmd.2011.5.1.

[18]

A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves,, preprint, (2012).

[19]

M. Yoshida, "Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory,", Aspects of Mathematics, E11 (1987).

[20]

A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

show all references

References:
[1]

F. Beukers, Gauss' hypergeometric functions,, , ().

[2]

______, Notes on differential equations and hypergeometric functions,, , ().

[3]

I. 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.

[4]

J. Carlson, S. Müller-Stach and C. Peters, "Period Mappings and Period Domains,", Cambridge Studies in Advanced Mathematics, 85 (2003).

[5]

T. A. Driscoll and L. N. Trefethen, "Schwarz-Christoffel Mapping,", Cambridge Monographs on Applied and Computational Mathematics, 8 (2002).

[6]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow,, preprint., ().

[7]

______, Lyapunov spectrum of square-tiled cyclic covers,, J. Mod. Dyn., 5 (2011), 319. doi: 10.3934/jmd.2011.5.319.

[8]

G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,, Ann. of Math. (2), 155 (2002), 1. doi: 10.2307/3062150.

[9]

______, On the Lyapunov exponents of the Kontsevich-Zorich cocycle,, in, (2006), 549.

[10]

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

[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.

[12]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic,, Duke Math. J., 103 (2000), 191. doi: 10.1215/S0012-7094-00-10321-3.

[13]

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

[14]

H. Masur and S. Tabachnikov, Rational billiards and flat structures,, in, (2002), 1015. doi: 10.1016/S1874-575X(02)80015-7.

[15]

C. Matheus and J.-C. Yoccoz, The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis,, Journal of Modern Dynamics, 4 (2010), 453. doi: 10.3934/jmd.2010.4.453.

[16]

M. Möller, Teichmüller curves, Galois actions and $\hat{GT}$-relations,, Math. Nachr., 278 (2005), 1061. doi: 10.1002/mana.200310292.

[17]

______, Shimura and Teichmüller curves,, J. Mod. Dyn., 5 (2011), 1. doi: 10.3934/jmd.2011.5.1.

[18]

A. Wright, Schwarz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves,, preprint, (2012).

[19]

M. Yoshida, "Fuchsian Differential Equations. With Special Emphasis on the Gauss-Schwarz Theory,", Aspects of Mathematics, E11 (1987).

[20]

A. Zorich, Flat surfaces,, in, (2006), 437. doi: 10.1007/978-3-540-31347-2_13.

[1]

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

[2]

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

[3]

Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1

[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]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048

[12]

Gabriel Fuhrmann, Jing Wang. Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5747-5761. doi: 10.3934/dcds.2017249

[13]

Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453

[14]

Klara Stokes, Maria Bras-Amorós. Associating a numerical semigroup to the triangle-free configurations. Advances in Mathematics of Communications, 2011, 5 (2) : 351-371. doi: 10.3934/amc.2011.5.351

[15]

Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165

[16]

Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Networks & Heterogeneous Media, 2010, 5 (3) : 487-505. doi: 10.3934/nhm.2010.5.487

[17]

John Banks. Topological mapping properties defined by digraphs. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83

[18]

Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315

[19]

Koray Karabina, Berkant Ustaoglu. Invalid-curve attacks on (hyper)elliptic curve cryptosystems. Advances in Mathematics of Communications, 2010, 4 (3) : 307-321. doi: 10.3934/amc.2010.4.307

[20]

Robert L. Devaney, Daniel M. Look. Buried Sierpinski curve Julia sets. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1035-1046. doi: 10.3934/dcds.2005.13.1035

2017 Impact Factor: 0.425

Metrics

  • PDF downloads (3)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]