Labeled Rauzy classes and framed translation surfaces

Boissy, Corentin

Annales de l'Institut Fourier, Tome 63 (2013) no. 2, pp. 547-572.

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.

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