Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-28T10:14:50.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Morphisms fixing words associated with exchange of three intervals

Published online by Cambridge University Press:  11 February 2010

Petr Ambrož ,
Zuzana Masáková  and
Edita Pelantová
Petr Ambrož
Affiliation:
Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected]; [email protected]; [email protected]
Zuzana Masáková
Affiliation:
Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected]; [email protected]; [email protected]
Edita Pelantová
Affiliation:
Doppler Institute & Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected]; [email protected]; [email protected]
Get access

Abstract

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.

Keywords

Interval exchange transformationSturmian morphismssubstitution invariance.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 44 , Issue 1: Special issue dedicated to the 12th "Journées Montoises d'Informatique Théorique" , January 2010 , pp. 3 - 17
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adamczewski, B., Codages de rotations et phénomènes d'autosimilarité. J. Théor. Nombres Bordeaux 14 (2002) 351–386. CrossRef
Allauzen, C., Une caractérisation simple des nombres de Sturm. J. Théor. Nombres Bordeaux 10 (1998) 237–241. CrossRef
Arnoux, P., Berthé, V., Masáková, Z. and Pelantová, E., Sturm numbers and substitution invariance of 3iet words. Integers 8 (2008) 17 (electronic).
Baláži, P., Masáková, Z. and Pelantová, E., Complete characterization of substitution invariant Sturmian sequences. Integers 5 (2005) 23 (electronic).
Baláži, P., Masáková, Z. and Pelantová, E., Characterization of substitution invariant 3iet words. Integers 8 (2008) 21 (electronic).
Berstel, J. and Séébold, P., Morphismes de Sturm. Bull. Belg. Math. Soc. Simon Stevin 1 (1994) 175–189. Journées Montoises (Mons, 1992).
Berthé, V., Ei, H., Ito, S. and Rao, H., On substitution invariant Sturmian words: an application of Rauzy fractals. RAIRO-Theor. Inf. Appl. 41 (2007) 329–349. CrossRef
Boshernitzan, M.D. and Carroll, C.R., An extension of Lagrange's theorem to interval exchange transformations over quadratic fields. J. Anal. Math. 72 (1997) 21–44. CrossRef
Crisp, D., Moran, W., Pollington, A. and Shiue, P., Substitution invariant cutting sequences. J. Théor. Nombres Bordeaux 5 (1993) 123–137. CrossRef
Ferenczi, S., Holton, C. and Zamboni, L.Q., Structure of three interval exchange transformations. I. An arithmetic study. Ann. Inst. Fourier 51 (2001) 861–901. CrossRef
Ferenczi, S., Holton, C. and Zamboni, L.Q., Structure of three-interval exchange transformations. II. A combinatorial description of the trajectories. J. Anal. Math. 89 (2003) 239–276. CrossRef
Ferenczi, S., Holton, C. and Zamboni, L.Q., Structure of three-interval exchange transformations III: ergodic and spectral properties. J. Anal. Math. 93 (2004) 103–138. CrossRef
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.
Komatsu, T. and van der Poorten, A.J., Substitution invariant Beatty sequences. Jpn J. Math. (N.S.) 22 (1996) 349–354.
Mignosi, F. and Séébold, P., Morphismes sturmiens et règles de Rauzy. J. Théor. Nombres Bordeaux 5 (1993) 221–233. CrossRef
Parvaix, B., Substitution invariant Sturmian bisequences. J. Théor. Nombres Bordeaux 11 (1999) 201–210. Les XXèmes Journées Arithmétiques (Limoges, 1997). CrossRef
M. Queffélec, Substitution dynamical systems–spectral analysis. Lect. Notes Math. 1294 (1987).
Séébold, P., Fibonacci morphisms and Sturmian words. Theoret. Comput. Sci. 88 (1991) 365–384. CrossRef
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.