Volume 7 (2024)
Boulanger, Julien; Lanneau, Erwan; Massart, Daniel
Algebraic intersection for a family of Veech surfaces
Annales Henri Lebesgue, Volume 7 (2024), pp. 787-821.

Metadata

DOI10.5802/ahl.211
Keywords Lyapunov exponents, translation surface, Teichmüller curve, algebraic intersection

Abstract

We study some properties of the function KVol defined by

KVol(X,ω):=Vol(X,ω)sup α,β Int(α,β) l g (α)l g (β)

on the moduli space of translation surfaces. For the Teichmüller discs 𝒯 n of the original Veech surfaces arising from the right-angled triangles (π/2,π/n,(n-2)π/2n) for odd n≥5, we establish the first known explicit formula for KVol (beyond the case of the moduli space of flat tori).


References

[Bou22] Boulanger, Julien Central points of the double heptagon translation surface are not connection points, Bull. Soc. Math. Fr., Volume 150 (2022) no. 2, pp. 459-472 | DOI | MR | Zbl

[Bou23a] Boulanger, Julien Algebraic intersection, lengths and Veech surfaces (2023) | arXiv

[Bou23b] Boulanger, Julien Quelques problèmes géométriques autour des surfaces de translation, Ph. D. Thesis, Université Grenoble-Alpes, Grenoble, France (2023)

[CKM21a] Cheboui, Smail; Kessi, Arezki; Massart, Daniel Algebraic intersection for translation surfaces in a family of Teichmüller disks, Bull. Soc. Math. Fr., Volume 149 (2021) no. 4, pp. 613-640 | DOI | MR | Zbl

[CKM21b] Cheboui, Smail; Kessi, Arezki; Massart, Daniel Algebraic intersection for translation surfaces in the stratum H(2), C. R. Math., Volume 359 (2021), pp. 65-70 | DOI | Numdam | MR | Zbl

[DFT11] Davis, Diana; Fuchs, Dmitry; Tabachnikov, Serge Periodic trajectories in the regular pentagon, Mosc. Math. J., Volume 11 (2011) no. 3, pp. 439-461 | Zbl

[DL18] Davis, Diana; Lelievre, Samuel Periodic paths on the pentagon, double pentagon and golden L (2018) | arXiv

[FL23] Freedman, Sam; Lucas, Trent Veech fibrations (2023) | arXiv

[Hoo13] Hooper, W. Patrick Grid graphs and lattice surfaces, Int. Math. Res. Not., Volume 2013 (2013) no. 12, pp. 2657-2698 | DOI | Zbl

[LN14] Lanneau, Erwan; Nguyen, Duc-Manh Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4, J. Topol., Volume 7 (2014) no. 2, pp. 475-522 | DOI | MR | Zbl

[Mas96] Massart, Daniel Normes stables des surfaces, Ph. D. Thesis, École Normale Supérieure de Lyon (1996)

[Mas22] Massart, Daniel A short introduction to translation surfaces, Veech surfaces, and Teichmüller dynamics, Surveys in geometry I, Springer, 2022, pp. 343-388 | DOI | MR | Zbl

[McM05] McMullen, Curtis T. Teichmüller curves in genus two: Discriminant and spin, Math. Ann., Volume 333 (2005) no. 1, pp. 87-130 | DOI | Zbl

[McM07] McMullen, Curtis T. Dynamics of SL 2 (ℝ) over moduli space in genus two, Ann. Math., Volume 165 (2007) no. 2, pp. 397-456 | DOI | Zbl

[MM14] Massart, Daniel; Muetzel, Bjoern On the intersection form of surfaces, Manuscr. Math., Volume 143 (2014) no. 1-2, pp. 19-49 | DOI | Zbl

[Mon05] Monteil, Thierry On the finite blocking property, Ann. Inst. Fourier, Volume 55 (2005) no. 4, pp. 1195-1217 | DOI | Numdam | Zbl

[SU10] Smillie, John; Ulcigrai, Corinna Geodesic flow on the Teichmüller disk of the regular octagon cutting sequences and octagon continued fractions maps, Dynamical numbers. Interplay between dynamical systems and number theory. A special program, May 1–July 31, 2009. International conference, MPI, Bonn, Germany, July 20–24, 2009. Proceedings (Contemporary Mathematics), Volume 532, American Mathematical Society, 2010, pp. 29-65 | Zbl

[SU11] Smillie, John; Ulcigrai, Corinna Beyond Sturmian sequences: coding linear trajectories in the regular octagon, Proc. Lond. Math. Soc., Volume 102 (2011) no. 2, pp. 291-340 | DOI | Zbl

[SW10] Smillie, John; Weiss, Barak Characterizations of lattice surfaces, Invent. Math., Volume 180 (2010) no. 3, pp. 535-557 | DOI | MR | Zbl

[Vee89] Veech, William A. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., Volume 97 (1989) no. 3, pp. 553-583 | DOI | Zbl

[Vor96] Vorobets, Ya. B. Planar structures and billiards in rational polygons, Russ. Math. Surv., Volume 51 (1996) no. 1, pp. 177-178 | DOI | Zbl

[Wri16] Wright, Alex From rational billiards to dynamics on moduli spaces, Bull. Am. Math. Soc., Volume 53 (2016) no. 1, pp. 41-56 | DOI | MR | Zbl

[Zor06] Zorich, Anton Flat surfaces, Frontiers in number theory, physics, and geometry. I On random matrices, zeta functions, and dynamical systems, Springer, 2006, pp. 437-583 | DOI | MR | Zbl