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Original Articles

Statistics of Square-Tiled Surfaces: Symmetry and Short Loops

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References

  • Boissy, C., Geninska, S. (2017). Systoles in translation surfaces. arXiv:1707.05060.
  • Delecroix, V., Fougeron, C., Lelievre, S. (2019). surface_dynamics—SageMath package, Version 0.4.1.
  • Delecroix, V., Goujard, E., Zograf, P., Zorich, A. (2019). Contribution of one-cylinder square-tiled surfaces to Masur-Veech volumes. arXiv:1903.10904.
  • Dozier, B. (2019). Equidistribution of saddle connections on translation surfaces. J. Mod. Dyn. 14: 87–120. doi:10.3934/jmd.2019004
  • Eskin, A., Masur, H. (2001). Asymptotic formulas on flat surfaces. Ergod. Theory Dyn. Syst. 21(2): 443–478. doi:10.1017/S0143385701001225
  • Eskin, A., Masur, H., Schmoll, M. (2003). Billiards in rectangles with barriers. Duke Math. J. 118(3): 427–463. doi:10.1215/S0012-7094-03-11832-3
  • Eskin, A., Okounkov, A. (2001). Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145(1): 59–103. doi:10.1007/s002220100142
  • Forester, M., Tang, R., Tao, J. (2018). Veech surfaces and simple closed curves. Isr. J. Math. 223(1): 323–342. doi:10.1007/s11856-017-1617-5
  • Forni, G., Matheus, C. (2008). An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum. arXiv:0810.0023.
  • Forni, G., Matheus, C., Zorich, A. (2011). Square-tiled cyclic covers. J. Mod. Dyn. 5(2): 285–318. doi:10.3934/jmd.2011.5.285
  • Herrlich, F., Schmithüsen, G. (2008). An extraordinary origami curve. Math. Nachr. 281(2): 219–237. doi:10.1002/mana.200510597
  • Hubert, P., Lelièvre, S. (2006). Prime arithmetic teichmüller discs in H(2) . Isr. J. Math. 151(1): 281–321. doi:10.1007/BF02777365
  • Hubert, P., Schmidt, T. A. (2006). An introduction to Veech surfaces. In: Katok, A., Hasselblatt, B. (Eds.) Handbook of Dynamical Systems, Vol. 1B. Amsterdam: Elsevier B. V., pp. 501–526.
  • Judge, C., Parlier, H. (2019). The maximum number of systoles for genus two Riemann surfaces with abelian differentials. Comment. Math. Helv. 94(2): 399–437. doi:10.4171/CMH/463
  • Lelièvre, S., Royer, E. (2006). Orbit countings in H(2) and quasimodular forms. Int. Math. Res. Notices. 2006(9): 42151.
  • Masur, H. (1982). Interval exchange transformations and measured foliations. Ann. Math. 115: 169–200. doi:10.2307/1971341
  • Masur, H. (1988). Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential. In: Drasin, D., Earle, C.J., Gehring, F.W., Kra, I., Marden, A. (Eds.) Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986). Mathematical Sciences Research Institute Publications, Vol. 10. New York: Springer, pp. 215–228.
  • Masur, H. (1990). The growth rate of trajectories of a quadratic differential. Ergod. Theory Dyn. Syst. 10(1): 151–176. doi:10.1017/S0143385700005459
  • McMullen, C. T. (2005). Teichmüller curves in genus two: discriminant and spin. Math. Ann. 333(1): 87–130. doi:10.1007/s00208-005-0666-y
  • Ramanujan, S. (2000). On certain arithmetical functions [Trans. Cambridge Philos. Soc. 22 (1916), no. 9, 159–184]. In: Collected Papers of Srinivasa Ramanujan. Providence, RI: AMS Chelsea Publishing, pp. 136–162.
  • Schmithüsen, G. (2004). An algorithm for finding the Veech group of an origami. Exp. Math. 13(4): 459–472. doi:10.1080/10586458.2004.10504555
  • Shrestha, S. T. (2018). Counting formulae for square-tiled surfaces in genus two. arXiv:1810.08687.
  • Smillie, J., Weiss, B. (2010). Characterizations of lattice surfaces. Invent. Math. 180(3): 535–557. doi:10.1007/s00222-010-0236-0
  • Veech, W. A. (1982). Gauss measures for transformations on the space of interval exchange maps. Ann. Math. 115(2): 201–242. doi:10.2307/1971391
  • Veech, W. A. (1989). Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3): 553–583. doi:10.1007/BF01388890
  • Zmiaikou, D. (2011). Origamis et groupes de permutation. PhD thesis. l’Université Paris-Sud 11.
  • Zorich, A. (2002). Square tiled surfaces and Teichmüller volumes of the moduli spaces of abelian differentials. In: Burger, M., Iozzi, A. (Eds.) Rigidity in Dynamics and Geometry (Cambridge, 2000). Berlin: Springer, pp. 459–471.
  • Zorich, A. (2006). Flat surfaces. In: Cartier, P.E., Julia, B., Moussa, P., Vanhove, P. (Eds.) Frontiers in Number Theory, Physics, and Geometry. I. Berlin: Springer, pp. 437–583.
  • Zorich, A. (2008). Explicit Jenkins-Strebel representatives of all strata of abelian and quadratic differentials. J. Mod. Dyn. 2(1): 139–185. doi:10.3934/jmd.2008.2.139

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