# American Institute of Mathematical Sciences

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

 1 Aix-Marseille Université LATP, case cour A, Faculté des Sciences de Saint Jerôme, Avenue Escadrille Normandie-Niemen, 13397 Marseille cedex 20, France

Received  January 2011 Revised  March 2012 Published  May 2012

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.
Citation: Corentin Boissy. Classification of Rauzy classes in the moduli space of Abelian and quadratic differentials. Discrete & Continuous Dynamical Systems - A, 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. [2] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. [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. doi: 10.4171/CMH/180. [4] C. Boissy, Degenerations of quadratic differentials on $\mathbb{CP}^1$,, Geometry and Topology, 12 (2008), 1345. 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). [6] C. Boissy and E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials,, Ergodic Theory Dynam. Systems, 29 (2009), 767. doi: 10.1017/S0143385708080565. [7] C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces,, Ann. Sci. École Norm. Sup. (4), 23 (1990), 469. [8] A. Douady and J. Hubbard, On the density of Strebel differentials,, Inventiones Math., 30 (1975), 175. 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. [10] J. Fickenscher, Self-inverses in Rauzy classes,, preprint, (2011). [11] A. Katok, Interval exchange transformations and some special flows are not mixing,, Israel J. Math., 35 (1980), 301. doi: 10.1007/BF02760655. [12] M. Keane, Interval exchange transformations,, Math. Zeit., 141 (1975), 25. 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. 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. [15] E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials,, Ann. Sci. École Norm. Sup. (4), 41 (2008), 1. [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. doi: 10.1090/S0894-0347-05-00490-X. [17] H. Masur, Interval exchange transformations and measured foliations,, Ann of Math. (2), 115 (1982), 169. 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. doi: 10.1007/BF02565820. [19] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, IN, (2002), 1015. [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. doi: 10.1007/s00039-008-0678-3. [21] G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. [22] W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. [23] W. Veech, Moduli spaces of quadratic differentials,, J. Analyse Math., 55 (1990), 117. 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.

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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. [2] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. [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. doi: 10.4171/CMH/180. [4] C. Boissy, Degenerations of quadratic differentials on $\mathbb{CP}^1$,, Geometry and Topology, 12 (2008), 1345. 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). [6] C. Boissy and E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials,, Ergodic Theory Dynam. Systems, 29 (2009), 767. doi: 10.1017/S0143385708080565. [7] C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces,, Ann. Sci. École Norm. Sup. (4), 23 (1990), 469. [8] A. Douady and J. Hubbard, On the density of Strebel differentials,, Inventiones Math., 30 (1975), 175. 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. [10] J. Fickenscher, Self-inverses in Rauzy classes,, preprint, (2011). [11] A. Katok, Interval exchange transformations and some special flows are not mixing,, Israel J. Math., 35 (1980), 301. doi: 10.1007/BF02760655. [12] M. Keane, Interval exchange transformations,, Math. Zeit., 141 (1975), 25. 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. 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. [15] E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials,, Ann. Sci. École Norm. Sup. (4), 41 (2008), 1. [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. doi: 10.1090/S0894-0347-05-00490-X. [17] H. Masur, Interval exchange transformations and measured foliations,, Ann of Math. (2), 115 (1982), 169. 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. doi: 10.1007/BF02565820. [19] H. Masur and S. Tabachnikov, Rational billiards and flat structures,, IN, (2002), 1015. [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. doi: 10.1007/s00039-008-0678-3. [21] G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. [22] W. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. [23] W. Veech, Moduli spaces of quadratic differentials,, J. Analyse Math., 55 (1990), 117. 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.
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