Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow
[Conditions de type Khinchin pour les surfaces de translation et lois asymptotiques pour le flot de Teichmüller]
Bulletin de la Société Mathématique de France, Tome 140 (2012) no. 4, pp. 485-532.

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.

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.

DOI : 10.24033/bsmf.2634
Classification : 37D40, 37A20, 11J70, 11K50
Keywords: translation surfaces, Teichmüller flow, Khinchin condition, interval exchange transformations
Mot clés : surfaces de translation, flot de Teichmüller, condition de Khinchin, transformations d'échange d'intervalles
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     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},
     pages = {485--532},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {140},
     number = {4},
     year = {2012},
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Marchese, Luca. Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow. Bulletin de la Société Mathématique de France, Tome 140 (2012) no. 4, pp. 485-532. doi : 10.24033/bsmf.2634. http://archive.numdam.org/articles/10.24033/bsmf.2634/

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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