Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials

Abstract

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.

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“…For µ hol , [22, Section 10, Claim (7)] proves the regularity above. A weaker regularity for any SL(2, R)-invariant measure is proved in [4,Theorem 1.2].…”

“…(4.4) µ(N ǫ,κ ) m 1 ǫκMasur and Smillie [22, Section 10, Claim(7)] show that the holonomy measure µ hol is regular.Avila, Matheus and Yoccoz [4, Theorem 1.2] prove a weaker regularity for any SL(2, R)-invariant measure. SL(2, R)-invariant loci, Siegel-Veech transform and volume asymptotic.…”

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“…Furthermore, since Q ;l is convex, the connected component of the stratum is uniquely determined by ( ; l/. This is discussed in detail in [2] .…”