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

Published online by Cambridge University Press:  01 June 2009

CORENTIN BOISSY  and
ERWAN LANNEAU
CORENTIN BOISSY
Affiliation:
IRMAR, Campus de Beaulieu, UMR CNRS 6625, Université de Rennes I, 35042 Rennes cedex, France (email: cboissy@latp.univ-mrs.fr)
ERWAN LANNEAU
Affiliation:
Centre de Physique Théorique (CPT), UMR CNRS 6207, Université du Sud Toulon-Var and Fédération de Recherches des Unités de Mathématiques de Marseille, Luminy, Case 907, F-13288 Marseille Cedex 9, France (email: lanneau@cpt.univ-mrs.fr)
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 3 , June 2009 , pp. 767 - 816
Copyright
Copyright © Cambridge University Press 2008

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