Dans cet article on considère la structure des projections des segments de coupure correspondant aux substitutions unimodulaires sur un alphabet binaire. On montre qu’une telle projection est un bloc de lettres si et seulement si la substitution est sturmienne. Une double application de ce procédé à une substitution de Christoffel donne la substitution originelle. On obtient ainsi une dualité sur l’ensemble des substitutions de Christoffel.
In this paper we study the structure of the projections of the finite cutting segments corresponding to unimodular substitutions over a two-letter alphabet. We show that such a projection is a block of letters if and only if the substitution is Sturmian. Applying the procedure of projecting the cutting segments corresponding to a Christoffel substitution twice results in the original substitution. This induces a duality on the set of Christoffel substitutions.
@article{JTNB_2007__19_2_523_0,
author = {Rosema, Sierk W.},
title = {Substitutions on two letters, cutting segments and their projections},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {523--545},
publisher = {Universit\'e Bordeaux 1},
volume = {19},
number = {2},
year = {2007},
doi = {10.5802/jtnb.600},
mrnumber = {2394900},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jtnb.600/}
}
TY - JOUR AU - Rosema, Sierk W. TI - Substitutions on two letters, cutting segments and their projections JO - Journal de théorie des nombres de Bordeaux PY - 2007 SP - 523 EP - 545 VL - 19 IS - 2 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.600/ DO - 10.5802/jtnb.600 LA - en ID - JTNB_2007__19_2_523_0 ER -
%0 Journal Article %A Rosema, Sierk W. %T Substitutions on two letters, cutting segments and their projections %J Journal de théorie des nombres de Bordeaux %D 2007 %P 523-545 %V 19 %N 2 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.600/ %R 10.5802/jtnb.600 %G en %F JTNB_2007__19_2_523_0
Rosema, Sierk W. Substitutions on two letters, cutting segments and their projections. Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 523-545. doi : 10.5802/jtnb.600. http://www.numdam.org/articles/10.5802/jtnb.600/
[1] J. Berstel, A. de Luca, Sturmian words, Lyndon words and trees. Theoret. Comput. Sci. 178 (1997), 171–203. | MR | Zbl
[2] E. B. Christoffel, Observatio arithmetica. Math. Ann. 6 (1875), 145–152.
[3] C. Fuchs, R. Tijdeman, Substitutions, abstract number systems and the space filling property. Ann. Inst. Fourier (Grenoble) 56 (2006), 2345–2389. | Numdam | MR
[4] M. Lothaire, Combinatorics on Words. Cambridge University Press, 1983. | MR | Zbl
[5] M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, 2002. | MR | Zbl
[6] M. Morse, G. A. Hedlund, Symbolic Dynamics. Amer. J. Math. 60 (1938), 815–866. | MR
[7] M. Morse, G. A. Hedlund, Symbolic Dynamics II: Sturmian trajectories. Amer. J. Math. 62 (1940), 1–42. | MR
[8] N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics. Springer, 2002. | MR
[9] G. Rauzy, Nombres algébriques et substitutions. Bull. Soc. Math. France 110 (1982), 147–178. | Numdam | MR | Zbl
[10] G. Richomme, Test-words for Sturmian morphisms. Bull. Belg. Math. Soc. 6 (1999), 481–489. | MR
[11] G. Richomme, Lyndon morphisms. Bull. Belg. Math. Soc. 10 (2003), 761–785. | MR | Zbl
[12] S. W. Rosema, R. Tijdeman, The tribonacci substitution. Integers: Electron. J. Combin. Number Th. 5(3) (2005), A13. | MR | Zbl
[13] P. Séébold, Fibonacci morphisms and Sturmian words. Theoret. Comput. Sci. 195 (1991), 91–109. | MR | Zbl
[14] C. Series, The geometry of Markoff numbers. Math. Intelligencer 7, no. 3 (1985), 20–29. | MR | Zbl
Cité par Sources :
