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

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October 2012, 32(10): 3433-3457. doi: 10.3934/dcds.2012.32.3433

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, France

Received January 2011 Revised March 2012 Published May 2012

Abstract
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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: Rauzy classes, Moduli spaces., Interval exchange maps, linear involutions, quadratic differentials.

Mathematics Subject Classification: Primary: 37E05; Secondary: 37D40, 32G1.

Citation: Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3433-3457. doi: 10.3934/dcds.2012.32.3433

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.

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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.

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