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A combinatorial approach to Rauzy-type dynamics II: The labelling method and a second proof of the KZB classification theorem
Université Paris-Saclay, France |
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [
However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up works.
Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method, we can prove most of the identities appearing in the literature so far ([
References:
[1] |
C. Boissy,
Classification of Rauzy classes in the moduli space of abelian and quadratic differentials, Discrete Contin. Dyn. Syst., 32 (2012), 3433-3457.
doi: 10.3934/dcds.2012.32.3433. |
[2] |
C. Boissy, Labeled Rauzy classes and framed translation surfaces, Annales de l'Institut Fourier, 63 (2013), 547–572.
doi: 10.5802/aif.2769. |
[3] |
C. Boissy and E. Lanneau,
Pseudo-Anosov homeomorphisms on translation surfaces in hyperelliptic components have large entropy, Geom. Funct. Anal., 22 (2012), 74-106.
doi: 10.1007/s00039-012-0152-0. |
[4] |
Q. de Mourgues, A Combinatorial Approach to Rauzy-Type Dynamics, Chapter 6, Ph.D thesis, Université Paris 13, 2017, https://hal.archives-ouvertes.fr/tel-02285651. |
[5] |
Q. de Mourgues and A. Sportiello, A combinatorial approach to Rauzy-type dynamics I: Permutations and the Kontsevich–Zorich–Boissy classification theorem,, https://arXiv.org/abs/1705.01641. |
[6] |
V. Delecroix,
Cardinalités des classes de Rauzy, Ann. Inst. Fourier, 63 (2013), 1651-1715.
doi: 10.5802/aif.2811. |
[7] |
A. Eskin, H. Masur and A. Zorich,
Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants, Publications Mathématiques de l'IHÉS, 97 (2003), 61-179.
doi: 10.1007/s10240-003-0015-1. |
[8] |
J. Fickenscher,
A combinatorial proof of the Kontsevich–Zorich–Boissy classification of {R}auzy classes, Discrete Contin. Dyn. Syst., 36 (2016), 1983-2025.
doi: 10.3934/dcds.2016.36.1983. |
[9] |
R. Gutiérrez-Romo, Classification of Rauzy-Veech groups: Proof of the Zorich conjecture, Invent. Math., 215 (2019), 741–778. eprint, arXiv: 1706.04923
doi: 10.1007/s00222-018-0836-7. |
[10] |
M. Kontsevich and A. Zorich,
Connected components of the moduli spaces of abelian differentials with prescribed singularities, Inventiones Mathematicæ, 153 (2003), 631-678.
doi: 10.1007/s00222-003-0303-x. |
[11] |
E. Lanneau,
Connected components of the strata of the moduli spaces of quadratic differentials, Annales Scientifiques de l'École Normale Supérieure, 41 (2008), 1-56.
doi: 10.24033/asens.2062. |
[12] |
G. Rauzy,
Échanges d'intervalles et transformations induites, Acta Arithmetica, 34 (1979), 315-328.
doi: 10.4064/aa-34-4-315-328. |
[13] |
W. A. Veech,
Gauss measures for transformations on the space of interval exchange maps, Ann. Math. (2), 115 (1982), 201-242.
doi: 10.2307/1971391. |
show all references
References:
[1] |
C. Boissy,
Classification of Rauzy classes in the moduli space of abelian and quadratic differentials, Discrete Contin. Dyn. Syst., 32 (2012), 3433-3457.
doi: 10.3934/dcds.2012.32.3433. |
[2] |
C. Boissy, Labeled Rauzy classes and framed translation surfaces, Annales de l'Institut Fourier, 63 (2013), 547–572.
doi: 10.5802/aif.2769. |
[3] |
C. Boissy and E. Lanneau,
Pseudo-Anosov homeomorphisms on translation surfaces in hyperelliptic components have large entropy, Geom. Funct. Anal., 22 (2012), 74-106.
doi: 10.1007/s00039-012-0152-0. |
[4] |
Q. de Mourgues, A Combinatorial Approach to Rauzy-Type Dynamics, Chapter 6, Ph.D thesis, Université Paris 13, 2017, https://hal.archives-ouvertes.fr/tel-02285651. |
[5] |
Q. de Mourgues and A. Sportiello, A combinatorial approach to Rauzy-type dynamics I: Permutations and the Kontsevich–Zorich–Boissy classification theorem,, https://arXiv.org/abs/1705.01641. |
[6] |
V. Delecroix,
Cardinalités des classes de Rauzy, Ann. Inst. Fourier, 63 (2013), 1651-1715.
doi: 10.5802/aif.2811. |
[7] |
A. Eskin, H. Masur and A. Zorich,
Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants, Publications Mathématiques de l'IHÉS, 97 (2003), 61-179.
doi: 10.1007/s10240-003-0015-1. |
[8] |
J. Fickenscher,
A combinatorial proof of the Kontsevich–Zorich–Boissy classification of {R}auzy classes, Discrete Contin. Dyn. Syst., 36 (2016), 1983-2025.
doi: 10.3934/dcds.2016.36.1983. |
[9] |
R. Gutiérrez-Romo, Classification of Rauzy-Veech groups: Proof of the Zorich conjecture, Invent. Math., 215 (2019), 741–778. eprint, arXiv: 1706.04923
doi: 10.1007/s00222-018-0836-7. |
[10] |
M. Kontsevich and A. Zorich,
Connected components of the moduli spaces of abelian differentials with prescribed singularities, Inventiones Mathematicæ, 153 (2003), 631-678.
doi: 10.1007/s00222-003-0303-x. |
[11] |
E. Lanneau,
Connected components of the strata of the moduli spaces of quadratic differentials, Annales Scientifiques de l'École Normale Supérieure, 41 (2008), 1-56.
doi: 10.24033/asens.2062. |
[12] |
G. Rauzy,
Échanges d'intervalles et transformations induites, Acta Arithmetica, 34 (1979), 315-328.
doi: 10.4064/aa-34-4-315-328. |
[13] |
W. A. Veech,
Gauss measures for transformations on the space of interval exchange maps, Ann. Math. (2), 115 (1982), 201-242.
doi: 10.2307/1971391. |




































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