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Computing periodic points on Veech surfaces

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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.

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Data availability statement

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


  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., 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). (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 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)

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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.

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Correspondence to Sam Freedman.

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Chowdhury, Z., Everett, S., Freedman, S. et al. Computing periodic points on Veech surfaces. Geom Dedicata 217, 66 (2023).

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