Skip to main content
SpringerLink
Log in

Computing periodic points on Veech surfaces

Geometriae Dedicata volume 217, Article number: 66 (2023) Cite this article

Abstract

A non-square-tiled Veech surface has finitely many periodic points, i.e. points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic points. Our algorithm serves as a new proof of the finiteness of periodic points for non-square-tiled Veech surfaces. We apply our algorithm to Prym eigenforms in the minimal stratum in genus 3, proving that in low discriminant these surfaces do not have periodic points, except for the fixed points of the Prym involution.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Data availability statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Apisa, P.: GL2R-invariant measures in marked strata: generic marked points, Earle-Kra for strata, and illumination. Geom. Topol. 24, 373–408 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Apisa, P., Saavedra, R.M., Zhang, C.: Periodic points on the regular and double n-gon surfaces (2020)

  3. Joshua, P.: Bowman, Teichmüller geodesics, Delaunay triangulations, and Veech groups. Ramanujan Math. Soc. Lect. Notes 10, 113–129 (2008)

    Google Scholar 

  4. Calta, K.: Veech surfaces and complete periodicity in genus two. J. Am. Math. Soc. 17, 871–908 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chowdhury, Z., Everett, S., Freedman, S., Lee, D.: Computing periodic points on Veech surfaces, GitHub. https://github.com/SFreedman67/bowman, 2021

  6. Eskin, A., Filip, S., Wright, A.: The algebraic hull of the Kontsevich–Zorich cocycle. Ann. Math. 188, 1–33 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Freedman, S.: Periodic points of Prym eigenforms, available at 2210.13503 (2022)

  8. Gutkin, E., Hubert, P.: Affine diffeomorphisms of translation surfaces: periodic points. Fuchsian groups arithmeticity 36(6), 847–866 (2003)

  9. Gutkin, E., Judge, C.: Affine mappings of translation surfaces: geometry and arithmetic. Duke Math. J. 103(2), 191–213 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hubert, P., Lanneau, E.: Veech groups without parabolic elements, Duke Math. J. 133 (June 2006), no. 2

  11. Hubert, P., Schmidt, T.A.: Chapter 6: An Introduction to Veech Surfaces, Handbook of Dynamical Systems, pp. 501–526 (2006)

  12. Katok, S.: Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics. Homog Flows Moduli Spaces Arith. 10, 243–320 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Lanneau, E., Nguyen, D.-M.: Teichmüller curves generated by Weierstrass Prym eigenforms in genus three and genus four. J. Topol. 7, 475–522 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lanneau, E., Nguyen, D.-M., Wright, A.: Finiteness of Teichmüller curves in non-arithmetic rank 1 orbit closures. Am. J. Math. 139(6), 1449–1463 (2017)

    Article  MATH  Google Scholar 

  15. Masur, H., Smillie, J.: Hausdorff dimension of sets of nonergodic measured foliations. Ann. Math. 134, 455–543 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Masur, H., Tabachnikov, S.: Rational Billiards and Flat Structures, Handbook of Dynamical Systems, vol. 1a, pp. 1015–1089 (2002)

  17. Curtis, T.: McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces. J. Am. Math. Soc. 16, 857–885 (2003)

    Article  MATH  Google Scholar 

  18. Curtis, T.: McMullen, Prym varieties and Teichmüller curves. Duke Math. J. 133, 569–590 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Möller, M.: Periodic points on Veech surfaces and the Mordell–Weil group over a Teichmüller curve 165(3), 633–649 (2006)

  20. Sage Developers, Sagemath, the Sage Mathematics Software System (Version 9.0). https://www.sagemath.org (2020)

  21. Shinomiya, Y.: Veech surfaces and their periodic points. Conform. Geom. Dyn. 20, 176–196 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Veech, W.A.: Bicuspid F-structures and Hecke groups. In: Proceedings of the London Mathematical Society 103 (201104), no. 4, 710–745. Available at https://academic.oup.com/ plms/article-pdf/103/4/710/4258173/pdq057.pdf

  23. Wright, A.: Translation surfaces and their orbit closures: an introduction for a broad audience. EMS Surv. Math. Sci. 2, 63–108 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wright, B.: Periodic points of Ward–Veech surfaces, available at 2106.09116 (2021)

  25. Zorich, A.: Flat surfaces, Frontiers in number theory, physics, and geometry vol. i, pp. 439–586 (2006)

Download references

Acknowledgements

We thank the Institute for Computational and Experimental Research in Mathematics (ICERM) for running the Summer@ICERM 2021 REU where this work took place. We also thank Curt McMullen for comments and suggestions on an early draft of this paper. Finally, we are grateful to Paul Apisa for proposing the problem and providing invaluable guidance and support.

Author information

Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, USA

    Zawad Chowdhury

  2. University of Chicago, Chicago, USA

    Samuel Everett

  3. Brown University, Providence, USA

    Sam Freedman

  4. Columbia University, New York, USA

    Destine Lee

Authors
  1. Zawad Chowdhury
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel Everett
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sam Freedman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Destine Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sam Freedman.

Ethics declarations

Conflict of interest

The authors have no conflicts of interests or competing interests. The authors have no financial or proprietary interests in any material discussed in this article. The authors have no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chowdhury, Z., Everett, S., Freedman, S. et al. Computing periodic points on Veech surfaces. Geom Dedicata 217, 66 (2023). https://doi.org/10.1007/s10711-023-00804-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10711-023-00804-z

Keywords

search

Navigation