Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials

Corentin Boissy

doi:10.3934/dcds.2012.32.3433

Article Contents

2012, Volume 32, Issue 10: 3433-3457. Doi: 10.3934/dcds.2012.32.3433

This issue Previous Article Inverting the Furstenberg correspondence Next Article Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system

Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials

Corentin Boissy^1,

1.
Aix-Marseille Université LATP, case cour A, Faculté des Sciences de Saint Jerôme, Avenue Escadrille Normandie-Niemen, 13397 Marseille cedex 20

Received: January 2011

Revised: March 2012

Published online: May 2012

Abstract / Introduction Related Papers Cited by

Abstract

We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a similar result for Rauzy classes corresponding to quadratic differentials.

Keywords:

Mathematics Subject Classification: Primary: 37E05; Secondary: 37D40, 32G15.

Citation:

Related Papers

Cited by

References

[1]	A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci., 104 (2006), 143-211.
[2]	A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture, Acta Math., 198 (2007), 1-56.
[3]	C. Boissy, Configurations of saddle connections of quadratic differentials on $\mathbb{CP}^1$ and on hyperelliptic Riemann surfaces, Comment. Math. Helv., 84 (2009), 757-791.doi: 10.4171/CMH/180.
[4]	C. Boissy, Degenerations of quadratic differentials on $\mathbb{CP}^1$, Geometry and Topology, 12 (2008), 1345-1386.doi: 10.2140/gt.2008.12.1345.
[5]	C. Boissy, Labeled Rauzy classes and framed translation surfaces, to appear in Annales de l'Institut Fourier, 2010, arXiv:1010.5719.
[6]	C. Boissy and E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials, Ergodic Theory Dynam. Systems, 29 (2009), 767-816.doi: 10.1017/S0143385708080565.
[7]	C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces, Ann. Sci. École Norm. Sup. (4), 23 (1990), 469-494.
[8]	A. Douady and J. Hubbard, On the density of Strebel differentials, Inventiones Math., 30 (1975), 175-179.doi: 10.1007/BF01425507.
[9]	A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Hautes Études Sci., 97 (2003), 61-179.
[10]	J. Fickenscher, Self-inverses in Rauzy classes, preprint, 2011, arXiv:1103.3485.
[11]	A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math., 35 (1980), 301-310.doi: 10.1007/BF02760655.
[12]	M. Keane, Interval exchange transformations, Math. Zeit., 141 (1975), 25-31.doi: 10.1007/BF01236981.
[13]	M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153 (2003), 631-678.doi: 10.1007/s00222-003-0303-x.
[14]	E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment. Math. Helv., 79 (2004), 471-501.
[15]	E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials, Ann. Sci. École Norm. Sup. (4), 41 (2008), 1-56.
[16]	S. Marmi, P. Moussa and J.-C. Yoccoz, The cohomological equation for Roth type interval exchange maps, Journal of the Amer. Math. Soc., 18 (2005), 823-872.doi: 10.1090/S0894-0347-05-00490-X.
[17]	H. Masur, Interval exchange transformations and measured foliations, Ann of Math. (2), 115 (1982), 169-200.doi: 10.2307/1971341.
[18]	H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov homeomorphisms, Comment. Math. Helv., 68 (1993), 289-307.doi: 10.1007/BF02565820.
[19]	H. Masur and S. Tabachnikov, Rational billiards and flat structures, IN "Handbook of Dynamical Systems," Vol. 1A, North-Holland, Amsterdam, (2002), 1015-1089.
[20]	H. Masur and A. Zorich, Multiple saddle connections on flat surfaces and the principal boundary of the moduli space of quadratic differentials, Geom. Funct. Anal., 18 (2008), 919-987.doi: 10.1007/s00039-008-0678-3.
[21]	G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.
[22]	W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242.doi: 10.2307/1971391.
[23]	W. Veech, Moduli spaces of quadratic differentials, J. Analyse Math., 55 (1990), 117-171.doi: 10.1007/BF02789200.
[24]	A. Zorich, Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials, Journal of Modern Dynamics, 2 (2008), 139-185.