A generalization of the self-dual induction to every interval exchange transformation

Ferenczi, Sébastien

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 1947-2002.

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.

Classification : 37B10, 68R15
Keywords: Dynamical systems, interval exchanges, symbolic dynamics
Mot clés : systèmes dynamiques, échanges d’intervalles, dynamique symbolique

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

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

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

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

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

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

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

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

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

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

[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, Volume 16 (1996) no. 4, pp. 663-682 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

[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, Volume 211 (1973), pp. 775-778 | MR | Zbl

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

[27] Keane, Michael Interval exchange transformations, Math. Z., Volume 141 (1975), pp. 25-31 | DOI | EuDML | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

[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), Volume 46 (1996) no. 2, pp. 325-370 | DOI | EuDML | Numdam | MR | Zbl

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