Interval exchanges, admissibility and branching Rauzy induction

Dolce, Francesco; Perrin, Dominique

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 51 (2017) no. 3, pp. 135-139.

We introduce a definition of admissibility for subintervals in interval exchange transformations. We characterize the admissible intervals using a branching version of the Rauzy induction. Using this notion, we prove a property of the natural codings of interval exchange transformations, namely that any derived set of a regular interval exchange set is a regular interval exchange set with the same number of intervals. Derivation is taken here with respect to return words. We also study the case of regular interval exchange transformations defined over a quadratic field and show that the set of factors of such a transformation is primitive morphic. The proof uses an extension of a result of Boshernitzan and Carroll.

Classification : 68R15, 37B10, 37E05
Mots clés : Interval exchange, Rauzy induction, return words, derived sets

V.I. Arnol’D, Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat. Nauk 18 (1963) 91–192. | MR | Zbl

P. Baláži, Z. Masáková and E. Pelantová, Characterization of substitution invariant words coding exchange of three intervals. Integers 8 (2008) A20, 21. | MR | Zbl

L’. Balková, E. Pelantová and W. Steiner, Sequences with constant number of return words. Monatsh. Math. 155 (2008) 251–263. | DOI | MR | Zbl

J. Berstel, C. De Felice, D. Perrin, C. Reutenauer and G. Rindone, Bifix codes and Sturmian words. J. Algebra 369 (2012) 146–202. | DOI | MR | Zbl

V. Berthé, C. De Felice, F. Dolce, J. Leroy, D. Perrin, C. Reutenauer and G. Rindone, Acyclic, connected and tree sets. Monatsh. Math (2015). | MR

V. Berthé, C. De Felice, F. Dolce, J. Leroy, D. Perrin, C. Reutenauer and G. Rindone, Maximal bifix decoding. Discrete Math. 338 (2015) 725–742. | DOI | MR | Zbl

V. Berthé and M. Rigo, Combinatorics, automata and number theory. Vol. 135 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2010). | MR | Zbl

V. Berthé, C. De Felice, F. Dolce and J. Leroy, Dominique Perrin, Christophe Reutenauer and Giuseppina Rindone. Bifix codes and interval exchanges. J. Pure Appl. Algebra 219 (2015) 2781–2798. | MR | Zbl

C. Boissy and E. Lanneau, Dynamics and geometry of the Rauzy-Veech induction for quadratic differentials. Ergodic Theory Dynam. Syst. 29 (2009) 767–816. | DOI | MR | Zbl

M.D. Boshernitzan, Rank two interval exchange transformations. Ergodic Theory Dynam. Syst. 8 (1988) 379–394. | DOI | MR | Zbl

M.D. Boshernitzan and C.R. Carroll, An extension of Lagrange’s theorem to interval exchange transformations over quadratic fields. J. Anal. Math. 72 (1997) 21–44. | DOI | MR | Zbl

I.P. Cornfeld, S.V. Fomin and Ya.G. Sinaĭ, Ergodic theory. Vol. 245 of Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences. Translated from the Russian by A.B. Sosinskiĭ. Springer Verlag, New York (1982). | MR | Zbl

C. Danthony and A. Nogueira, Measured foliations on nonorientable surfaces. Ann. Sci. École Norm. Sup. 23 (1990) 469–494. | DOI | Numdam | MR | Zbl

F. Durand, A characterization of substitutive sequences using return words. Discrete Math. 179 (1998) 89–101. | DOI | MR | Zbl

S. Ferenczi and L.Q. Zamboni, Languages of $k$-interval exchange transformations. Bull. Lond. Math. Soc. 40 (2008) 705–714. | DOI | MR | Zbl

N. Pytheas Fogg, Substitutions in dynamics, arithmetics and combinatorics. Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel. Vol. 1794 of Lect. Notes Math. Springer-Verlag, Berlin (2002). | MR | Zbl

Y. Jullian, An algorithm to identify automorphisms which arise from self-induced interval exchange transformations. Math. Z. 274 (2013) 33–55. | DOI | MR | Zbl

J. Justin and L. Vuillon, Return words in Sturmian and episturmian words. Theoret. Inform. Appl. 34 (2000) 343–356. | DOI | Numdam | MR | Zbl

M. Keane, Interval exchange transformations. Math. Z. 141 (1975) 25–31. | DOI | MR | Zbl

P. Køurka, Topological and symbolic dynamics. Vol. 11 of Cours Spécialisés, Specialized Courses. Société Mathématique de France, Paris (2003). | MR | Zbl

M. Lothaire, Algebraic combinatorics on words. Vol. 90 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2002). | Zbl

T. Miernowski and A. Nogueira, Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations. Ergodic Theory Dynam. Syst. 33 (2013) 221–246. | DOI | MR | Zbl

V.I. Oseledec, The spectrum of ergodic automorphisms. Dokl. Akad. Nauk SSSR 168 (1966) 1009–1011. | MR | Zbl

G. Rauzy, Échanges d’intervalles et transformations induites. Acta Arith. 34 (1979) 315–328. | DOI | MR | Zbl

A. Skripchenko, Symmetric interval identification systems of order three. Discrete Contin. Dyn. Syst. 32 (2012) 643–656. | DOI | MR | Zbl

M. Viana, Ergodic theory of interval exchange maps. Rev. Mat. Complut. 19 (2006) 7–100. | DOI | MR | Zbl

L. Vuillon, On the number of return words in infinite words constructed by interval exchange transformations. Pure Math. Appl. (PU.M.A.) 18 (2007) 345–355. | MR | Zbl

J.-C. Yoccoz, Interval exchange maps and translation surfaces. In Homogeneous flows, moduli spaces and arithmetic. Vol. 10 of Clay Math. Proc. Amer. Math. Soc., Providence, RI (2010) 1–69. | MR | Zbl