no. 17-18
Combinatorics
On the triplex substitution – Rauzy fractal
[Sur la substitution triplexe – fractal de Rauzy]

Présenté par : Jean-Pierre Kahane

Tan, Bo1 ; Wen, Zhi-Xiong1 ; Zhang, Yiping2
1 Department of Mathematics, Huazhong University of Science and Technology, 430074, Wuhan, PR China
2 Department of Mathematics, Wuhan University, Wuhan 430072, PR China
Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 925-930.

On étudie le fractal de Rauzy engendré par la substitution triplexe σ:a↦ab, b↦acb, c↦acc. Analysant la structure combinatoire des préfixes, qui est différente de celle des substitutions de Pisot, nous caractérisons les points du fractal de Rauzy en termes des nombres de Fibonacci. En particulier, des propriétés lexicographiques et de similarité locale sont données.

This Note is the continuation of our previous one. Here we study the Rauzy fractal associated with the triplex substitution σ:a↦ab, b↦acb, c↦acc. Analyzing the combinatorial structure of the prefixes, we characterize the set of points in the Rauzy fractal in terms of Fibonacci numbers, which is quite different from the case of Pisot substitutions. Some lexicographic properties and local similarity structures are presented.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.06.014
Tan, Bo 1 ; Wen, Zhi-Xiong 1 ; Zhang, Yiping 2

1 Department of Mathematics, Huazhong University of Science and Technology, 430074, Wuhan, PR China
2 Department of Mathematics, Wuhan University, Wuhan 430072, PR China 
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Tan, Bo; Wen, Zhi-Xiong; Zhang, Yiping. On the triplex substitution – Rauzy fractal. Comptes Rendus. Mathématique, Tome 346 (2008) no. 17-18, pp. 925-930. doi : 10.1016/j.crma.2008.06.014. http://www.numdam.org/articles/10.1016/j.crma.2008.06.014/

[1] Arnoux, P.; Ito, S. Pisot substitutions and Rauzy fractals, Marne-la-Vallée, 2000 (Bull. Belg. Math. Soc. Simon Stevin), Volume 8 (2001), pp. 181-207

[2] Ei, H.; Ito, S.; Rao, H. Atomic surfaces, tilings and coincidences II. Reducible case, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 7, pp. 2285-2313

[3] Pytheas Fogg, N. Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002

[4] Rauzy, G. Nombres algébriques et substitutions, Bull. Soc. Math. France, Volume 110 (1982) no. 2, pp. 147-178

[5] B. Tan, Z.-X. Wen, Y. Zhang, On the triplex substitution – combinatorial properties C. R. Acad. Sci. Paris, Ser. I 346 (2008), in press, | DOI

[6] Wen, Z.-Y.; Janssen, T.; Dekking, F.M. Fibonacci chain as a periodic chain with discommensurations, J. Phys. A, Volume 27 (1994) no. 5, pp. 1691-1702

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⁎ Research supported by NSFC No. 10501035, 10631040, 10571140 and 10671150.