Substitutions with Cofinal Fixed Points
[Substitutions de Points fixes co-finaux]
TAN, Bo ; WEN, Zhi-Xiong ; WU, Jun ; WEN, Zhi-Ying
Annales de l'Institut Fourier, Tome 56 (2006), p. 2551-2563 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Soit ϕ une substitution en un alphabet {a,b} de deux lettres. Si ϕ(a) et ϕ(b) commencent par a et b respectivement, alors ϕ possède deux points fixes débutants par a et b respectivement.

Nous caractériserons les substitutions avec deux points fixes co-finaux (c’est-à-dire, qui diffèrent que par leur préfixe). La démonstration est combinatoire, elle se base sur une étude de répétitions de mots dans les points fixes.

Let ϕ be a substitution over a 2-letter alphabet, say {a,b}. If ϕ(a) and ϕ(b) begin with a and b respectively, ϕ has two fixed points beginning with a and b respectively.

We characterize substitutions with two cofinal fixed points (i.e., which differ only by prefixes). The proof is a combinatorial one, based on the study of repetitions of words in the fixed points.

Publié le : 2006-01-01
DOI : https://doi.org/10.5802/aif.2249
Classification:  68R15,  11B85
Mots clés: Suites co-finales, substitution 
@article{AIF_2006__56_7_2551_0,
     author = {TAN, Bo and WEN, Zhi-Xiong and WU, Jun and WEN, Zhi-Ying},
     title = {Substitutions with Cofinal Fixed Points},
     journal = {Annales de l'Institut Fourier},
     volume = {56},
     year = {2006},
     pages = {2551-2563},
     doi = {10.5802/aif.2249},
     zbl = {1121.68092},
     mrnumber = {2290790},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2006__56_7_2551_0}
}

TAN, Bo; WEN, Zhi-Xiong; WU, Jun; WEN, Zhi-Ying. Substitutions with Cofinal Fixed Points. Annales de l'Institut Fourier, Tome 56 (2006) pp. 2551-2563. doi : 10.5802/aif.2249. http://gdmltest.u-ga.fr/item/AIF_2006__56_7_2551_0/

[1] Allouche, J.P.; Shallit, J.O. Automatic sequences: Theory and Applications, Cambridge University Press, Cambrige (2002) | Zbl 01993704

[2] Arnoux, P.; Rauzy, G. Représentation géométrique de suites de complexité 2n+1, Bull. Soc. Math., Tome 119 (1991) no. 2, pp. 199-215 (France) | Numdam | MR 1116845 | Zbl 0789.28011

[3] Arnoux, P.; Ito, S Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin, Tome 8 (2001) no. 2, pp. 181-207 | MR 1838930 | Zbl 1007.37001

[4] Ei, H.; Ito, S. Decomposition theorem on invertible substitutions, Osaka J. Math., Tome 35 (1998) no. 4, pp. 821-834 | MR 1659624 | Zbl 0924.20040

[5] Lothaire, M. Combinatorics on words, Cambridge University Press, Cambridge (1997) | MR 1475463 | Zbl 0874.20040

[6] Lothaire, M. Algebraic combinatorics on words, Cambridge University Press, Cambridge (2002) | MR 1905123 | Zbl 1001.68093

[7] Lothaire, M. Applied combinatorics on words, Cambridge University Press, Cambridge (2005) | MR 2165687 | Zbl 02183071

[8] Nielsen, J. Die Isomorphismengruppen der freien Gruppen, Math. Ann, Tome 91 (1924), pp. 169-209 (Available at http://mathlib.sub.uni-goettingen.de/JFM/digit.php?an=JFM+50.0078.04) | Article | MR 1512188

[9] Pytheas Fogg, N. Substitutions in Dynamics, Arithmetics and Combinatorics, Springer, Berlin, Lecture Notes in Mathematics, Tome 1794 (2002) | MR 1970385 | Zbl 1014.11015

[10] Séébold, P. An effective solution to the D0L periodicity problem in the binary case, EATCS Bull., Tome 36 (1988), pp. 137-151 | Zbl 0678.68072

[11] Tan, B.; Wen, Z.-X.; Zhang, Y. P. The structure of invertible substitutions on a three-letter alphabet, Adv. in Appl. Math., Tome 32 (2004) no. 4, pp. 736-753 | Article | MR 2053843 | Zbl 1082.68092

[12] Wen, Z.-X.; Wen, Z.-Y. Local isomorphism of the invertible substitutions, C. R. Acad. Sci. Paris Sér. I Math., Tome 318 (1994) no. 4, pp. 299-304 | MR 1267604 | Zbl 0812.11018

[13] Wen, Z.-X.; Wen, Z.-Y.; Wu, J. On invertible substitutions with two fixed points, C. R. Math. Acad. Sci. Paris, Tome 334 (2002) no. 9, pp. 727-731 | MR 1905029 | Zbl 0996.68149

[14] Wen, Z.-X.; Zhang, Y. P. Some remarks on invertible substitutions on three letter alphabet, Chinese Sci. Bull., Tome 44 (1999) no. 19, pp. 1755-1760 | Article | MR 1737516 | Zbl 1040.20504