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

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

1. 

Fine Hall - Washington Road, Princeton, NJ 08544-1000, United States

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.  Google Scholar

[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.  Google Scholar

[3]

C. Boissy, Labeled rauzy classes and framed translation surfaces,, Annales de L'Institut Fourier, 63 (2013), 547.  doi: 10.5802/aif.2769.  Google Scholar

[4]

C. Boissy, A combinatorial move on the set of jenkins-strebel differentials,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

V. Delecroix, Cardinality of Rauzy classes,, Annales de l'institute Fourier, 63 (2013), 1651.  doi: 10.5802/aif.2811.  Google Scholar

[8]

J. Fickenscher, Self-inverses in Rauzy Classes,, Ph.D thesis, (2011).   Google Scholar

[9]

J. Fickenscher, Labeled and non-labeled extended Rauzy classes,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315.   Google Scholar

[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.  Google Scholar

[15]

W. A. Veech, Moduli spaces of quadratic differentials,, J. Analyse Math., 55 (1990), 117.  doi: 10.1007/BF02789200.  Google Scholar

[16]

M. Viana, Ergodic theory of interval exchange maps,, Rev. Mat. Complut., 19 (2006), 7.  doi: 10.5209/rev_rema.2006.v19.n1.16621.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

C. Boissy, Labeled rauzy classes and framed translation surfaces,, Annales de L'Institut Fourier, 63 (2013), 547.  doi: 10.5802/aif.2769.  Google Scholar

[4]

C. Boissy, A combinatorial move on the set of jenkins-strebel differentials,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[7]

V. Delecroix, Cardinality of Rauzy classes,, Annales de l'institute Fourier, 63 (2013), 1651.  doi: 10.5802/aif.2811.  Google Scholar

[8]

J. Fickenscher, Self-inverses in Rauzy Classes,, Ph.D thesis, (2011).   Google Scholar

[9]

J. Fickenscher, Labeled and non-labeled extended Rauzy classes,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

G. Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315.   Google Scholar

[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.  Google Scholar

[15]

W. A. Veech, Moduli spaces of quadratic differentials,, J. Analyse Math., 55 (1990), 117.  doi: 10.1007/BF02789200.  Google Scholar

[16]

M. Viana, Ergodic theory of interval exchange maps,, Rev. Mat. Complut., 19 (2006), 7.  doi: 10.5209/rev_rema.2006.v19.n1.16621.  Google Scholar

[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.  Google Scholar

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