Morphisms fixing words associated with exchange of three intervals. (English) Zbl 1186.68342

Summary: We consider words coding exchange of three intervals with permutation (3,2,1), here called 3iet words. Recently, a characterization of substitution invariant 3iet words was provided. We study the opposite question: what are the morphisms fixing a 3iet word? We reveal a narrow connection of such morphisms and morphisms fixing Sturmian words using the new notion of amicability.


68R15 Combinatorics on words
08A50 Word problems (aspects of algebraic structures)
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[1] B. Adamczewski, Codages de rotations et phénomènes d’autosimilarité. J. Théor. Nombres Bordeaux14 (2002) 351-386. Zbl1113.37003 · Zbl 1113.37003
[2] C. Allauzen, Une caractérisation simple des nombres de Sturm. J. Théor. Nombres Bordeaux10 (1998) 237-241. · Zbl 0930.11051
[3] P. Arnoux, V. Berthé, Z. Masáková and E. Pelantová, Sturm numbers and substitution invariance of 3iet words. Integers8 (2008) 17 (electronic). Zbl1202.11021 · Zbl 1202.11021
[4] P. Baláži, Z. Masáková and E. Pelantová, Complete characterization of substitution invariant Sturmian sequences. Integers5 (2005) 23 (electronic). Zbl1121.11020 · Zbl 1121.11020
[5] P. Baláži, Z. Masáková and E. Pelantová, Characterization of substitution invariant 3iet words. Integers8 (2008) 21 (electronic). · Zbl 1210.68078
[6] J. Berstel and P. Séébold, Morphismes de Sturm. Bull. Belg. Math. Soc. Simon Stevin1 (1994) 175-189. Journées Montoises (Mons, 1992).
[7] V. Berthé, H. Ei, S. Ito and H. Rao, On substitution invariant Sturmian words: an application of Rauzy fractals. RAIRO-Theor. Inf. Appl.41 (2007) 329-349. Zbl1140.11014 · Zbl 1140.11014
[8] 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. · Zbl 0931.28013
[9] D. Crisp, W. Moran, A. Pollington and P. Shiue, Substitution invariant cutting sequences. J. Théor. Nombres Bordeaux5 (1993) 123-137. Zbl0786.11041 · Zbl 0786.11041
[10] S. Ferenczi, C. Holton and L.Q. Zamboni, Structure of three interval exchange transformations. I. An arithmetic study. Ann. Inst. Fourier51 (2001) 861-901. Zbl1029.11036 · Zbl 1029.11036
[11] S. Ferenczi, C. Holton and L.Q. Zamboni, Structure of three-interval exchange transformations. II. A combinatorial description of the trajectories. J. Anal. Math.89 (2003) 239-276. Zbl1130.37324 · Zbl 1130.37324
[12] S. Ferenczi, C. Holton and L.Q. Zamboni, Structure of three-interval exchange transformations III: ergodic and spectral properties. J. Anal. Math.93 (2004) 103-138. · Zbl 1094.37005
[13] M. Fiedler, Special matrices and their applications in numerical mathematics. Martinus Nijhoff Publishers, Dordrecht (1986). Translated from the Czech by Petr Přikryl and Karel Segeth. · Zbl 0677.65019
[14] T. Komatsu and A.J. van der Poorten, Substitution invariant Beatty sequences. Jpn J. Math. (N.S.)22 (1996) 349-354. · Zbl 0868.11015
[15] F. Mignosi and P. Séébold, Morphismes sturmiens et règles de Rauzy. J. Théor. Nombres Bordeaux5 (1993) 221-233. · Zbl 0797.11029
[16] B. Parvaix, Substitution invariant Sturmian bisequences. J. Théor. Nombres Bordeaux11 (1999) 201-210. Les XXèmes Journées Arithmétiques (Limoges, 1997). Zbl0978.11005 · Zbl 0978.11005
[17] M. Queffélec, Substitution dynamical systems-spectral analysis. Lect. Notes Math. 1294 (1987). · Zbl 0642.28013
[18] P. Séébold, Fibonacci morphisms and Sturmian words. Theoret. Comput. Sci.88 (1991) 365-384. · Zbl 0737.68068
[19] S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions. In Number theory and its applications (Kyoto, 1997), Dev. Math.2, Kluwer Acad. Publ. (1999) 347-373. · Zbl 0971.11007
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