A generalization of the self-dual induction to every interval exchange transformation
[Une généralisation de l’induction autoduale à tous les échanges d’intervalles]
Ferenczi, Sébastien
Annales de l'Institut Fourier, Tome 64 (2014), p. 1947-2002 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous généralisons à tous les échanges d’intervalles l’algorithme d’induction défini par Ferenczi et Zamboni pour une classe particulière. Chaque échange d’intervalles correspond à un chemin infini dans un graphe dont les sommets sont certaines unions d’arbres que nous appelons des forêts de châteaux. Nous l’utilisons pour décrire les mots obtenus en codant les trajectoires, et donner une représentation explicite du système par des tours de Rokhlin. Comme application, nous construisons le premier exemple connu d’échange d’intervalles faiblement mélangeant en-dehors de la classe de Rauzy hyper-elliptique et de celle des rotations.

We generalize to all interval exchanges the induction algorithm defined by Ferenczi and Zamboni for a particular class. Each interval exchange corresponds to an infinite path in a graph whose vertices are certain unions of trees we call castle forests. We use it to describe those words obtained by coding trajectories and give an explicit representation of the system by Rokhlin towers. As an application, we build the first known example of a weakly mixing interval exchange outside the hyperelliptic and rotations Rauzy classes.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/aif.2901
Classification:  37B10,  68R15
Mots clés: systèmes dynamiques, échanges d’intervalles, dynamique symbolique 
@article{AIF_2014__64_5_1947_0,
     author = {Ferenczi, S\'ebastien},
     title = {A generalization of the self-dual induction to every interval exchange transformation},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {1947-2002},
     doi = {10.5802/aif.2901},
     zbl = {06387328},
     mrnumber = {3330928},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_5_1947_0}
}

Ferenczi, Sébastien. A generalization of the self-dual induction to every interval exchange transformation. Annales de l'Institut Fourier, Tome 64 (2014) pp. 1947-2002. doi : 10.5802/aif.2901. http://gdmltest.u-ga.fr/item/AIF_2014__64_5_1947_0/

[1] Adamczewski, Boris; Bugeaud, Yann Transcendence and Diophantine approximation, Combinatorics, automata and number theory, Cambridge Univ. Press, Cambridge (Encyclopedia Math. Appl.) Tome 135 (2010), pp. 410-451 | MR 2759111 | Zbl 1271.11073

[2] ArnolʼD, V. I. Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, Tome 18 (1963) no. 6 (114), pp. 91-192 | MR 170705 | Zbl 0135.42701

[3] Avila, Artur; Forni, Giovanni Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2), Tome 165 (2007) no. 2, pp. 637-664 | Article | MR 2299743 | Zbl 1136.37003

[4] Boshernitzan, Michael A unique ergodicity of minimal symbolic flows with linear block growth, J. Analyse Math., Tome 44 (1984/85), pp. 77-96 | Article | MR 801288 | Zbl 0602.28008

[5] Boshernitzan, Michael A condition for minimal interval exchange maps to be uniquely ergodic, Duke Math. J., Tome 52 (1985) no. 3, pp. 723-752 | Article | MR 808101 | Zbl 0602.28009

[6] Boshernitzan, Michael A condition for weak mixing of induced IETs, Dynamical systems and group actions, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 567 (2012), pp. 53-65 | MR 2931909 | Zbl 1262.37006

[7] Bourgain, J. On the correlation of the Moebius function with rank-one systems, J. Anal. Math., Tome 120 (2013), pp. 105-130 | Article | MR 3095150

[8] Cruz, Simone D.; Da Rocha, Luiz Fernando C. A generalization of the Gauss map and some classical theorems on continued fractions, Nonlinearity, Tome 18 (2005) no. 2, pp. 505-525 | Article | MR 2122671 | Zbl 1153.11320

[9] Delecroix, Vincent Cardinality of Rauzy classes, Ann. Inst. Fourier (Grenoble), Tome 63 (2013) no. 5, pp. 1651-1715 | Article | Numdam | MR 3186505 | Zbl 1285.05007

[10] Delecroix, Vincent Divergent trajectories in the periodic wind-tree model, J. Mod. Dyn., Tome 7 (2013) no. 1, pp. 1-29 | Article | MR 3071463 | Zbl 1291.37048

[11] Delecroix, Vincent; Ulcigrai, C. Diagonal changes for surfaces in hyperelliptic components (to appear in Geometriae Dedicata, arXiv:1310.1052)

[12] Ferenczi, Sébastien Diagonal changes for every interval exchange transformation (preprint, http://iml.univ-mrs.fr/~ferenczi/fid.pdf)

[13] Ferenczi, Sébastien Rank and symbolic complexity, Ergodic Theory Dynam. Systems, Tome 16 (1996) no. 4, pp. 663-682 | Article | MR 1406427 | Zbl 0858.68051

[14] Ferenczi, Sébastien Systems of finite rank, Colloq. Math., Tome 73 (1997) no. 1, pp. 35-65 | MR 1436950 | Zbl 0883.28014

[15] Ferenczi, Sébastien Billiards in regular 2n-gons and the self-dual induction, J. Lond. Math. Soc. (2), Tome 87 (2013) no. 3, pp. 766-784 | Article | MR 3073675 | Zbl 1283.37018

[16] Ferenczi, Sébastien Combinatorial methods for interval exchange transformations, Southeast Asian Bull. Math., Tome 37 (2013) no. 1, pp. 47-66 | MR 3100024 | Zbl 1289.37006

[17] Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q. Structure of three interval exchange transformations. I. An arithmetic study, Ann. Inst. Fourier (Grenoble), Tome 51 (2001) no. 4, pp. 861-901 | Article | Numdam | MR 1849209 | Zbl 1029.11036

[18] Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q. Structure of three-interval exchange transformations. II. A combinatorial description of the trajectories, J. Anal. Math., Tome 89 (2003), pp. 239-276 | Article | MR 1981920 | Zbl 1130.37324

[19] Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q. Structure of three-interval exchange transformations III: ergodic and spectral properties, J. Anal. Math., Tome 93 (2004), pp. 103-138 | Article | MR 2110326 | Zbl 1094.37005

[20] Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q. Structure of K-interval exchange transformations: induction, trajectories, and distance theorems, J. Anal. Math., Tome 112 (2010), pp. 289-328 | Article | MR 2763003 | Zbl 1225.37003

[21] Ferenczi, Sébastien; Da Rocha, Luiz Fernando C. A self-dual induction for three-interval exchange transformations, Dyn. Syst., Tome 24 (2009) no. 3, pp. 393-412 | Article | MR 2561448 | Zbl 1230.37005

[22] Ferenczi, Sébastien; Zamboni, Luca Q. Languages of k-interval exchange transformations, Bull. Lond. Math. Soc., Tome 40 (2008) no. 4, pp. 705-714 | Article | MR 2441143 | Zbl 1147.37008

[23] Ferenczi, Sébastien; Zamboni, Luca Q. Eigenvalues and simplicity of interval exchange transformations, Ann. Sci. Éc. Norm. Supér. (4), Tome 44 (2011) no. 3, pp. 361-392 | Numdam | MR 2839454 | Zbl 1237.37010

[24] Inoue, K.; Nakada, H. On the dual of Rauzy induction (preprint)

[25] Katok, A. B. Invariant measures of flows on orientable surfaces, Dokl. Akad. Nauk SSSR, Tome 211 (1973), pp. 775-778 | MR 331438 | Zbl 0298.28013

[26] Katok, A. B.; Stepin, A. M. Approximations in ergodic theory, Uspehi Mat. Nauk, Tome 22 (1967) no. 5 (137), pp. 81-106 (translated in Russian Math. Surveys 22 (1967), p. 76–102) | MR 219697 | Zbl 0172.07202

[27] Keane, Michael Interval exchange transformations, Math. Z., Tome 141 (1975), pp. 25-31 | Article | MR 357739 | Zbl 0278.28010

[28] Keane, Michael Non-ergodic interval exchange transformations, Israel J. Math., Tome 26 (1977) no. 2, pp. 188-196 | Article | MR 435353 | Zbl 0351.28012

[29] Kontsevich, Maxim; Zorich, Anton Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Tome 153 (2003) no. 3, pp. 631-678 | Article | MR 2000471 | Zbl 1087.32010

[30] Lopes, Artur O.; Da Rocha, Luiz Fernando C. Invariant measures for Gauss maps associated with interval exchange maps, Indiana Univ. Math. J., Tome 43 (1994) no. 4, pp. 1399-1438 | Article | MR 1322625 | Zbl 0840.28007

[31] Marmi, S.; Moussa, P.; Yoccoz, J.-C. The cohomological equation for Roth-type interval exchange maps, J. Amer. Math. Soc., Tome 18 (2005) no. 4, p. 823-872 (electronic) | Article | MR 2163864 | Zbl 1112.37002

[32] Masur, Howard Interval exchange transformations and measured foliations, Ann. of Math. (2), Tome 115 (1982) no. 1, pp. 169-200 | Article | MR 644018 | Zbl 0497.28012

[33] Oseledec, V. I. The spectrum of ergodic automorphisms, Dokl. Akad. Nauk SSSR, Tome 168 (1966), pp. 1009-1011 | MR 199347 | Zbl 0152.33404

[34] Penner, R. C.; Harer, J. L. Combinatorics of train tracks, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 125 (1992), pp. xii+216 | MR 1144770 | Zbl 0765.57001

[35] Rauzy, Gérard Échanges d’intervalles et transformations induites, Acta Arith., Tome 34 (1979) no. 4, pp. 315-328 | MR 543205 | Zbl 0414.28018

[36] Schweiger, Fritz Ergodic theory of fibred systems and metric number theory, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1995), pp. xiv+295 | MR 1419320 | Zbl 0819.11027

[37] Sinai, Ya. G.; Ulcigrai, C. Weak mixing in interval exchange transformations of periodic type, Lett. Math. Phys., Tome 74 (2005) no. 2, pp. 111-133 | Article | MR 2191950 | Zbl 1105.37002

[38] Veech, William A. Interval exchange transformations, J. Analyse Math., Tome 33 (1978), pp. 222-272 | Article | MR 516048 | Zbl 0455.28006

[39] Veech, William A. Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), Tome 115 (1982) no. 1, pp. 201-242 | Article | MR 644019 | Zbl 0486.28014

[40] Vershik, A. M.; Livshits, A. N. Adic models of ergodic transformations, spectral theory, substitutions, and related topics, Representation theory and dynamical systems, Amer. Math. Soc., Providence, RI (Adv. Soviet Math.) Tome 9 (1992), pp. 185-204 | MR 1166202 | Zbl 0770.28013

[41] Viana, M. Dynamics of interval exchange maps and Teichmüller flows (preliminary manuscript available from http://w3.impa.br/~viana/out/ietf.pdf)

[42] Yoccoz, Jean-Christophe Échanges d’intervalles (2005) (Cours au Collège de France, available from http://www.college-de-france.fr/site/jean-christophe-yoccoz/)

[43] Zorich, Anton Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), Tome 46 (1996) no. 2, pp. 325-370 | Article | Numdam | MR 1393518 | Zbl 0853.28007

[44] Zorich, Anton Explicit Jenkins-Strebel representatives of all strata of abelian and quadratic differentials, J. Mod. Dyn., Tome 2 (2008) no. 1, pp. 139-185 | Article | MR 2366233 | Zbl 1149.30033