# American Institute of Mathematical Sciences

2016, 36(4): 1983-2025. doi: 10.3934/dcds.2016.36.1983

## A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes

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Received  November 2014 Revised  July 2015 Published  September 2015

Rauzy Classes and Extended Rauzy Classes are equivalence classes of permutations that arise when studying Interval Exchange Transformations. In 2003, Kontsevich and Zorich classified Extended Rauzy Classes by using data from Translation Surfaces, which are associated to IET's thanks to the Zippered Rectangle Construction of Veech from 1982. In 2009, Boissy finalized the classification of Rauzy Classes also using information from Translation Surfaces. We present in this paper specialized moves in (Extended) Rauzy Classes that allow us to prove the sufficiency and necessity in the previous classification theorems. These results provide a complete, and purely combinatorial, proof of these known results. We end with some general statements about our constructed move.
Citation: Jon Fickenscher. A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1983-2025. doi: 10.3934/dcds.2016.36.1983
##### References:
 [1] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1. [2] C. Boissy, Classification of Rauzy classes in the moduli space of quadratic differentials,, Discrete and Continuous Dynam. Systems - A, 32 (2012), 3433. doi: 10.3934/dcds.2012.32.3433. [3] C. Boissy, Labeled rauzy classes and framed translation surfaces,, Annales de L'Institut Fourier, 63 (2013), 547. doi: 10.5802/aif.2769. [4] C. Boissy, A combinatorial move on the set of jenkins-strebel differentials,, preprint, (). [5] 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. [6] D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, Annales scientifiques de l'École normale supérieure, 47 (2014), 309. [7] V. Delecroix, Cardinality of Rauzy classes,, Annales de l'institute Fourier, 63 (2013), 1651. doi: 10.5802/aif.2811. [8] J. Fickenscher, Self-inverses in Rauzy Classes,, Ph.D thesis, (2011). [9] J. Fickenscher, Labeled and non-labeled extended Rauzy classes,, preprint, (). [10] J. Fickenscher, Self-inverses, Lagrangian permutations and minimal interval exchange transformations with many ergodic measures,, Comm. in Contemporary Mathematics, 16 (2014). doi: 10.1142/s0219199713500193. [11] 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. [12] E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials,, Annales scientifiques de l'École normale supérieure, 41 (2008), 1. [13] G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. [14] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. [15] W. A. Veech, Moduli spaces of quadratic differentials,, J. Analyse Math., 55 (1990), 117. doi: 10.1007/BF02789200. [16] M. Viana, Ergodic theory of interval exchange maps,, Rev. Mat. Complut., 19 (2006), 7. doi: 10.5209/rev_rema.2006.v19.n1.16621. [17] A. Zorich, Explicit Jenkins-Strebel representatives of all strata of abelian and quadratic differentials., J. Mod. Dyn., 2 (2008), 139. doi: 10.3934/jmd.2008.2.139.

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##### References:
 [1] A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,, Acta Math., 198 (2007), 1. doi: 10.1007/s11511-007-0012-1. [2] C. Boissy, Classification of Rauzy classes in the moduli space of quadratic differentials,, Discrete and Continuous Dynam. Systems - A, 32 (2012), 3433. doi: 10.3934/dcds.2012.32.3433. [3] C. Boissy, Labeled rauzy classes and framed translation surfaces,, Annales de L'Institut Fourier, 63 (2013), 547. doi: 10.5802/aif.2769. [4] C. Boissy, A combinatorial move on the set of jenkins-strebel differentials,, preprint, (). [5] 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. [6] D. Chen and M. Möller, Quadratic differentials in low genus: Exceptional and non-varying,, Annales scientifiques de l'École normale supérieure, 47 (2014), 309. [7] V. Delecroix, Cardinality of Rauzy classes,, Annales de l'institute Fourier, 63 (2013), 1651. doi: 10.5802/aif.2811. [8] J. Fickenscher, Self-inverses in Rauzy Classes,, Ph.D thesis, (2011). [9] J. Fickenscher, Labeled and non-labeled extended Rauzy classes,, preprint, (). [10] J. Fickenscher, Self-inverses, Lagrangian permutations and minimal interval exchange transformations with many ergodic measures,, Comm. in Contemporary Mathematics, 16 (2014). doi: 10.1142/s0219199713500193. [11] 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. [12] E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials,, Annales scientifiques de l'École normale supérieure, 41 (2008), 1. [13] G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315. [14] W. A. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201. doi: 10.2307/1971391. [15] W. A. Veech, Moduli spaces of quadratic differentials,, J. Analyse Math., 55 (1990), 117. doi: 10.1007/BF02789200. [16] M. Viana, Ergodic theory of interval exchange maps,, Rev. Mat. Complut., 19 (2006), 7. doi: 10.5209/rev_rema.2006.v19.n1.16621. [17] A. Zorich, Explicit Jenkins-Strebel representatives of all strata of abelian and quadratic differentials., J. Mod. Dyn., 2 (2008), 139. doi: 10.3934/jmd.2008.2.139.
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