Labeled Rauzy classes and framed translation surfaces

Boissy, Corentin

Annales de l'Institut Fourier, Tome 65 (2015) no. 2, pp. 905-932.

Dans cet article, on compare deux définitions de classes de Rauzy. La première a été introduite par Rauzy et a été utilisée en particulier par Veech pour démonter l’ergodicité du flot de Teichmüller. La seconde est plus récente et utilise un «  étiquetage  » des intervalles sous-jacents. Elle a été utilisée récemment dans les preuves de plusieurs résultats majeurs sur le flot de Teichmüller.

Les diagrammes de Rauzy obtenus avec la seconde définition sont des revêtements de ceux obtenus avec la première définition. On donne ici une formule donnant le degré de ce revêtement.

Cette formule est reliée à un espace des modules de surfaces de translations marquées, qui correspond à des surfaces de translations pour lesquelles on marque des séparatrices horizontales sur la surface. On calcule le nombre de composantes connexes de ces revêtements naturels de l’espace des modules des surfaces de translation.

Delecroix a donné récemment le cardinal des classes de Rauzy (réduites). On peut donc en déduire le cardinal des classes de Rauzy marquées.

In this paper, we compare two definitions of Rauzy classes. The first one was introduced by Rauzy and was in particular used by Veech to prove the ergodicity of the Teichmüller flow. The second one is more recent and uses a “labeling” of the underlying intervals, and was used in the proof of some recent major results about the Teichmüller flow.

The Rauzy diagrams obtained from the second definition are coverings of the initial ones. In this paper, we give a formula that gives the degree of this covering.

This formula is related to moduli spaces of framed translation surfaces, which correspond to surfaces where we label horizontal separatrices on the surface. We compute the number of connected component of these natural coverings of the moduli spaces of translation surfaces.

Delecroix has given recently a formula for the cardinality of the (reduced) Rauzy classes. Therefore, we also obtain formula for labeled Rauzy classes.

[1] Avila, A.; Gouëzel, S.; Yoccoz, J.-C. Exponential mixing for the Teichmüller flow, Publ. Math. IHES, Volume 104 (2006), pp. 143-211 | DOI | Numdam | MR | Zbl

[2] Avila, A.; Viana, M. Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture, Acta Math., Volume 198 (2007) no. 1, pp. 1-56 | DOI | MR | Zbl

[3] Boissy, C. Degenerations of quadratic differentials on ${\mathrm{ℂ\mathbb{P}}}^1$, Geometry and Topology, Volume 12 (2008), pp. 1345-1386 | DOI | MR | Zbl

[4] Boissy, C. Classification of Rauzy classes in the moduli space of abelian and quadratic differentials, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 10, pp. 3433-3457 | DOI | MR | Zbl

[5] Boissy, C.; Lanneau, E. Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials, Ergodic Theory Dynam. Systems, Volume 29 (2009) no. 3, pp. 767-816 | DOI | MR | Zbl

[6] Boissy, C.; Lanneau, E. Pseudo-Anosov homeomorphisms on translation surfaces in hyperelliptic components have large entropy, Geom. Funct. Anal., Volume 22 (2012) no. 1, pp. 74-106 | DOI | MR | Zbl

[7] Bufetov, A. I. Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials, J. Amer. Math. Soc., Volume 19 (2006) no. 3, pp. 579-623 | DOI | MR | Zbl

[8] Danthony, C.; Nogueira, A. Measured foliations on nonorientable surfaces, Ann. Sci. École Norm. Sup. (4), Volume 23 (1990), pp. 469-494 | Numdam | MR | Zbl

[9] Delecroix, V. Cardinality of Rauzy classes, Ann. Inst. Fourier (Grenoble), Volume 63 (2013) no. 5, pp. 1651-1715 | DOI | Numdam | MR | Zbl

[10] Eskin, A.; Masur, H.; Zorich, A. Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants, Publ. Math. IHES, Volume 97 (2003), pp. 61-179 | DOI | Numdam | MR | Zbl

[11] Hubbard, J.; Masur, H. Quadratic differentials and foliations, Acta Math., Volume 142 (1979), pp. 221-274 | DOI | MR | Zbl

[12] Kerckhoff, S. P. Simplicial systems for interval exchange maps and measured foliations, Ergodic Theory Dynam. Systems, Volume 5 (1985) no. 2, pp. 257-271 | DOI | MR | Zbl

[13] Kontsevich, M.; Zorich, A. Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Volume 153 (2003) no. 3, pp. 631-678 | DOI | MR | Zbl

[14] Marchese, L. Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow, Bull. Soc. Math. France, Volume 140 (2012) no. 4, p. 485-532 (2013) | Numdam | MR | Zbl

[15] Marmi, S.; Moussa, P.; Yoccoz, J.-C. The cohomological equation for Roth type interval exchange transformations, Journal of the Amer. Math. Soc., Volume 18 (2005), pp. 823-872 | DOI | MR | Zbl

[16] Masur, H. Interval exchange transformations and measured foliations, Ann of Math., Volume 141 (1982), pp. 169-200 | DOI | MR | Zbl

[17] Rauzy, G. Échanges d’intervalles et transformations induites, Acta Arith., Volume 34 (1979), pp. 315-328 | MR | Zbl

[18] Stein, W. A. et al Sage Mathematics Software (Version 4.2.1), The Sage Development Team, 2009 (http://www.sagemath.org)

[19] Veech, W. Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), Volume 115 (1982) no. 1, pp. 201-242 | DOI | MR | Zbl