Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow
Bulletin de la Société Mathématique de France, Volume 140 (2012) no. 4, p. 485-532

We study a diophantine property for translation surfaces, defined in terms of saddle connections and inspired by classical Khinchin condition. We prove that the same dichotomy holds as in Khinchin theorem, then we deduce a sharp estimate on how fast the typical Teichmüller geodesic wanders towards infinity in the moduli space of translation surfaces. Finally we prove some stronger result in genus one.

On étude une propriété diophantienne pour les surfaces de translation, définie en termes de connexions de selles et inspirée par la condition de Khinchin classique. On prouve la même dichotomie du théorème de Khinchin et on en déduit une estimation sur la vitesse des excursions à l'infini pour une géodésique de Teichmüller typique dans l'espace des modules des surfaces de translation. Enfin on preuve un résultat plus fort en genre un.

DOI : https://doi.org/10.24033/bsmf.2634
Classification:  37D40,  37A20,  11J70,  11K50
Keywords: translation surfaces, Teichmüller flow, Khinchin condition, interval exchange transformations
@article{BSMF_2012__140_4_485_0,
     author = {Marchese, Luca},
     title = {Khinchin type condition for translation surfaces and asymptotic laws for the Teichm\"uller flow},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {140},
     number = {4},
     year = {2012},
     pages = {485-532},
     doi = {10.24033/bsmf.2634},
     zbl = {1268.37033},
     mrnumber = {3059848},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2012__140_4_485_0}
}
Marchese, Luca. Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow. Bulletin de la Société Mathématique de France, Volume 140 (2012) no. 4, pp. 485-532. doi : 10.24033/bsmf.2634. http://www.numdam.org/item/BSMF_2012__140_4_485_0/

[1] C. Boissy - « Labeled Rauzy classes and framed translation surfaces », preprint arXiv:1010.5719. | Numdam | MR 3112841 | Zbl pre06193040

[2] C. Boissy & E. Lanneau - « Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials », Ergodic Theory Dynam. Systems 29 (2009), p. 767-816. | MR 2505317 | Zbl 1195.37030

[3] C. Danthony & A. Nogueira - « Involutions linéaires et feuilletages mesurés », C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), p. 409-412. | MR 965808 | Zbl 0672.57016

[4] A. Eskin & H. Masur - « Asymptotic formulas on flat surfaces », Ergodic Theory Dynam. Systems 21 (2001), p. 443-478. | MR 1827113 | Zbl 1096.37501

[5] A. Y. Khintchine - Continued fractions, Translated by Peter Wynn, P. Noordhoff Ltd., 1963. | JFM 63.0924.02 | MR 161834 | Zbl 0117.28503

[6] M. Kontsevich & A. Zorich - « Connected components of the moduli spaces of Abelian differentials with prescribed singularities », Invent. Math. 153 (2003), p. 631-678. | MR 2000471 | Zbl 1087.32010

[7] L. Marchese - « The Khinchin theorem for interval-exchange transformations », J. Mod. Dyn. 5 (2011), p. 123-183. | MR 2787600 | Zbl 1219.37005

[8] S. Marmi, P. Moussa & J.-C. Yoccoz - « The cohomological equation for Roth-type interval exchange maps », J. Amer. Math. Soc. 18 (2005), p. 823-872. | MR 2163864 | Zbl 1112.37002

[9] H. Masur - « Interval exchange transformations and measured foliations », Ann. of Math. 115 (1982), p. 169-200. | MR 644018 | Zbl 0497.28012

[10] -, « Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential », in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., vol. 10, Springer, 1988, p. 215-228. | MR 955824 | Zbl 0661.30034

[11] -, « Logarithmic law for geodesics in moduli space », in Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991), Contemp. Math., vol. 150, Amer. Math. Soc., 1993, p. 229-245. | MR 1234267 | Zbl 0790.32022

[12] C. Series - « The modular surface and continued fractions », J. London Math. Soc. 31 (1985), p. 69-80. | MR 810563 | Zbl 0545.30001

[13] D. Sullivan - « Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics », Acta Math. 149 (1982), p. 215-237. | MR 688349 | Zbl 0517.58028

[14] W. A. Veech - « Gauss measures for transformations on the space of interval exchange maps », Ann. of Math. 115 (1982), p. 201-242. | MR 644019 | Zbl 0486.28014

[15] J.-C. Yoccoz - « Petits diviseurs en dimension 1 », Astérisque 231 (1994). | Zbl 0836.30001

[16] -, « Interval exchange maps and translation surfaces », lecture notes of the CMI summer school course, Centro di ricerca matematica Ennio de Giorgi, Pisa, June-July 2007.

[17] A. Zorich - « Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents », Ann. Inst. Fourier (Grenoble) 46 (1996), p. 325-370. | Numdam | MR 1393518 | Zbl 0853.28007