American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Möbius disjointness for interval exchange transformations on three intervals
  • JMD Home
  • This Volume
  • Next Article
    Siegel–Veech transforms are in $ \boldsymbol{L^2} $(with an appendix by Jayadev S. Athreya and Rene Rühr)
  2019, 14: 21-54. doi: 10.3934/jmd.2019002

The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces

Artur Avila 1, , Carlos Matheus 2, and  Jean-Christophe Yoccoz 3,

1. 

IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
2. 

Centre de Mathématiques Laurent Schwartz, CNRS (UMR 7640), École Polytechnique, 91128 Palaiseau, France
3. 

Collège de France, 3 Rue d'Ulm, Paris, Cedex 05, France

Received  December 20, 2017 Revised  July 12, 2018 Published  March 2019

We describe the Kontsevich–Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich–Zorich monodromies of $ SU(p,q) $ type are realized by appropriate covering constructions.

Keywords: Teichmüller dynamics, cyclic covers, Veech-McMullen families, hyperelliptic Rauzy diagrams, complex reflection group.
Mathematics Subject Classification: Primary: 37D40; Secondary: 37C85.
Citation: Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002
References:
[1]

A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy–Veech groups, Math. Ann., 370 (2018), 785-809. doi: 10.1007/s00208-017-1568-5. Google Scholar

[2]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich–Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1. Google Scholar

[3]

S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle, Duke Math. J., 166 (2017), 657-706. doi: 10.1215/00127094-3715806. Google Scholar

[4]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar

[5]

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. Google Scholar

[6]

C. Matheus, M. Möller and J.-C. Yoccoz, A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces, Invent. Math., 202 (2015), 333-425. doi: 10.1007/s00222-014-0565-5. Google Scholar

[7]

C. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946. doi: 10.1007/s00208-012-0804-2. Google Scholar

[8]

J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar

[9]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890. Google Scholar

show all references

References:
[1]

A. Avila, C. Matheus and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy–Veech groups, Math. Ann., 370 (2018), 785-809. doi: 10.1007/s00208-017-1568-5. Google Scholar

[2]

A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich–Kontsevich conjecture, Acta Math., 198 (2007), 1-56. doi: 10.1007/s11511-007-0012-1. Google Scholar

[3]

S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle, Duke Math. J., 166 (2017), 657-706. doi: 10.1215/00127094-3715806. Google Scholar

[4]

E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J., 103 (2000), 191-213. doi: 10.1215/S0012-7094-00-10321-3. Google Scholar

[5]

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. Google Scholar

[6]

C. Matheus, M. Möller and J.-C. Yoccoz, A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces, Invent. Math., 202 (2015), 333-425. doi: 10.1007/s00222-014-0565-5. Google Scholar

[7]

C. McMullen, Braid groups and Hodge theory, Math. Ann., 355 (2013), 893-946. doi: 10.1007/s00208-012-0804-2. Google Scholar

[8]

J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Google Scholar

[9]

W. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890. Google Scholar

[1]

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
[2]

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
[3]

Martin Möller. Shimura and Teichmüller curves. Journal of Modern Dynamics, 2011, 5 (1) : 1-32. doi: 10.3934/jmd.2011.5.1
[4]

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
[5]

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
[6]

Giovanni Forni, Carlos Matheus, Anton Zorich. Square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 285-318. doi: 10.3934/jmd.2011.5.285
[7]

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
[8]

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
[9]

Alex Eskin, Maxim Kontsevich, Anton Zorich. Lyapunov spectrum of square-tiled cyclic covers. Journal of Modern Dynamics, 2011, 5 (2) : 319-353. doi: 10.3934/jmd.2011.5.319
[10]

Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[11]

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
[12]

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
[13]

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
[14]

E. Kapsza, Gy. Károlyi, S. Kovács, G. Domokos. Regular and random patterns in complex bifurcation diagrams. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 519-540. doi: 10.3934/dcdsb.2003.3.519
[15]

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
[16]

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
[17]

Hao Wang, Yang Kuang. Alternative models for cyclic lemming dynamics. Mathematical Biosciences & Engineering, 2007, 4 (1) : 85-99. doi: 10.3934/mbe.2007.4.85
[18]

Rodrigo Abarzúa, Nicolas Thériault, Roberto Avanzi, Ismael Soto, Miguel Alfaro. Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates. Advances in Mathematics of Communications, 2010, 4 (2) : 115-139. doi: 10.3934/amc.2010.4.115
[19]

Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3301-3314. doi: 10.3934/dcdsb.2016098
[20]

Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2941-2954. doi: 10.3934/dcdsb.2018293

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (36)
  • HTML views (252)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences