Labeled Rauzy classes and framed translation surfaces

Corentin Boissy[1]

  • [1] Aix-Marseille Université, LATP, case cour A, Faculté de Saint Jérôme Avenue Escadrille Normandie-Niemen, 13397 Marseille cedex 20

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 2, page 547-572
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Boissy, Corentin. "Labeled Rauzy classes and framed translation surfaces." Annales de l’institut Fourier 63.2 (2013): 547-572. <http://eudml.org/doc/275517>.

@article{Boissy2013,
abstract = {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.},
affiliation = {Aix-Marseille Université, LATP, case cour A, Faculté de Saint Jérôme Avenue Escadrille Normandie-Niemen, 13397 Marseille cedex 20},
author = {Boissy, Corentin},
journal = {Annales de l’institut Fourier},
keywords = {Interval exchange maps; Rauzy induction; Abelian differentials; Moduli spaces; Teichmüller flow; interval exchange maps; moduli spaces},
language = {eng},
number = {2},
pages = {547-572},
publisher = {Association des Annales de l’institut Fourier},
title = {Labeled Rauzy classes and framed translation surfaces},
url = {http://eudml.org/doc/275517},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Boissy, Corentin
TI - Labeled Rauzy classes and framed translation surfaces
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 2
SP - 547
EP - 572
AB - 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.
LA - eng
KW - Interval exchange maps; Rauzy induction; Abelian differentials; Moduli spaces; Teichmüller flow; interval exchange maps; moduli spaces
UR - http://eudml.org/doc/275517
ER -

References

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