Abstract. In two fundamental classical papers, Masur [Ma1] and Veech [Ve1] have independently proved that the Teichmüller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work [KoZo], Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials.Here we are interested in the complementary case. In a previous paper [La1], we have described some particular component, namely the hyperelliptic connected components, and showed that some strata are non-connected. In this paper, we give the general classification theorem: up to four exceptional cases in low genera, the strata of meromorphic quadratic differentials are either connected, or have exactly two connected components. In this last case, one component is hyperelliptic, the other not. This result was announced in the paper [La1].Our proof is based on a new approach of the so-called Jenkins-Strebel differential. We will present and use the notion of generalized permutations. Notre preuve repose principalement sur une nouvelle approche des différentielles quadratiques de type Jenkins-Strebelà savoir la notion de permutations généralisées. Résumé

Link to this article: http://journals.cambridge.org/abstract_S0143385708080565How to cite this article: CORENTIN BOISSY and ERWAN LANNEAU (2009). Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials.Abstract. Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy-Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with Z/2Z linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy-Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

This paper is devoted to the classification of the infinite families of Teichmüller curves generated by Prym eigenforms in genus 3 (and partially in genus 4) having a single zero (previously discovered by . Our classification involves two invariants: the discriminant D and the generators of the corresponding quadratic order. By definition D might have values 0, 1, 4, 5 mod 8. It turns out that for D sufficiently large, there are two Teichmüller curves when D ≡ 1 mod 8, only one Teichmüller curve when D ≡ 0, 4 mod 8, and no Teichmüller curves when D ≡ 5 mod 8. For small values of D the number of Teichmüller curves is given by an explicit list. The ingredients of our proof are first, a description of these curves in terms of prototypes and models, and then a careful analysis of the combinatorial connectedness in the spirit of McMullen's paper [Math. Ann. 333 (2005) 87-130]. As a corollary we obtain a description of cusps of Teichmüller curves given by Prym eigenforms.We would like also to emphasize that even though we have the same statement compared to McMullen's work, when D ≡ 1 mod 8, the reason for this disconnectedness is different.