American Institute of Mathematical Sciences

Advanced
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

Jon Fickenscher 1,

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.
Keywords: Interval exchange transformation, translation surfaces., Rauzy induction.
Mathematics Subject Classification: Primary: 37A05; Secondary: 30F30, 32G15, 37E3.
Citation: Jon Fickenscher. A combinatorial proof of the Kontsevich-Zorich-Boissy classification of Rauzy classes. Discrete & Continuous Dynamical Systems, 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), 1-56. 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), 3433-3457. 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), 547-572. 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), 767-816. 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), 309-369.  Google Scholar

[7]

V. Delecroix, Cardinality of Rauzy classes, Annales de l'institute Fourier, 63 (2013), 1651-1715. doi: 10.5802/aif.2811.  Google Scholar

[8]

J. Fickenscher, Self-inverses in Rauzy Classes, Ph.D thesis, Rice University, 2011, arXiv:1103.3485.  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), 1350019, 51pp. 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), 631-678. 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), 1-56.  Google Scholar

[13]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

[14]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar

[15]

W. A. Veech, Moduli spaces of quadratic differentials, J. Analyse Math., 55 (1990), 117-171. doi: 10.1007/BF02789200.  Google Scholar

[16]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. 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), 139-185. 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), 1-56. 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), 3433-3457. 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), 547-572. 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), 767-816. 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), 309-369.  Google Scholar

[7]

V. Delecroix, Cardinality of Rauzy classes, Annales de l'institute Fourier, 63 (2013), 1651-1715. doi: 10.5802/aif.2811.  Google Scholar

[8]

J. Fickenscher, Self-inverses in Rauzy Classes, Ph.D thesis, Rice University, 2011, arXiv:1103.3485.  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), 1350019, 51pp. 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), 631-678. 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), 1-56.  Google Scholar

[13]

G. Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.  Google Scholar

[14]

W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.  Google Scholar

[15]

W. A. Veech, Moduli spaces of quadratic differentials, J. Analyse Math., 55 (1990), 117-171. doi: 10.1007/BF02789200.  Google Scholar

[16]

M. Viana, Ergodic theory of interval exchange maps, Rev. Mat. Complut., 19 (2006), 7-100. 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), 139-185. doi: 10.3934/jmd.2008.2.139.  Google Scholar

[1]

Sébastien Labbé. Rauzy induction of polygon partitions and toral $ \mathbb{Z}^2 $-rotations. Journal of Modern Dynamics, 2021, 17: 481-528. doi: 10.3934/jmd.2021017
[2]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
[3]

Giovanni Forni, Carlos Matheus. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. Journal of Modern Dynamics, 2014, 8 (3&4) : 271-436. doi: 10.3934/jmd.2014.8.271
[4]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004
[5]

Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[6]

Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715
[7]

Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[8]

Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006
[9]

Luca Marchese. The Khinchin Theorem for interval-exchange transformations. Journal of Modern Dynamics, 2011, 5 (1) : 123-183. doi: 10.3934/jmd.2011.5.123
[10]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[11]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
[12]

Alexander I. Bufetov. Hölder cocycles and ergodic integrals for translation flows on flat surfaces. Electronic Research Announcements, 2010, 17: 34-42. doi: 10.3934/era.2010.17.34
[13]

Artur Avila, Carlos Matheus, Jean-Christophe Yoccoz. The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces. Journal of Modern Dynamics, 2019, 14: 21-54. doi: 10.3934/jmd.2019002
[14]

Eugene Gutkin. Insecure configurations in lattice translation surfaces, with applications to polygonal billiards. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 367-382. doi: 10.3934/dcds.2006.16.367
[15]

Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307
[16]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
[17]

Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253
[18]

Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53
[19]

Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[20]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy