On the triplex substitution – Rauzy fractal

Bo Tan; Zhi-Xiong Wen; Yiping Zhang

doi:10.1016/j.crma.2008.06.014

Combinatorics

On the triplex substitution – Rauzy fractal
[Sur la substitution triplexe – fractal de Rauzy]

Présenté par : Jean-Pierre Kahane

Bo Tan ¹ ; Zhi-Xiong Wen ¹ ; Yiping Zhang 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

Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 925-930.

Résumé
Abstract

On étudie le fractal de Rauzy engendré par la substitution triplexe $σ : a \mapsto a b$ , $b \mapsto a c b$ , $c \mapsto a c c$ . 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 $σ : a \mapsto a b$ , $b \mapsto a c b$ , $c \mapsto a c c$ . 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 : 2008-02-18
Accepté le : 2008-06-18
Publié le : 2008-07-25

DOI : 10.1016/j.crma.2008.06.014

Affiliations des auteurs :

Bo Tan ¹ ; Zhi-Xiong Wen ¹ ; Yiping Zhang 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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     author = {Bo Tan and Zhi-Xiong Wen and Yiping Zhang},
     title = {On the triplex substitution {\textendash} {Rauzy} fractal},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {925--930},
     publisher = {Elsevier},
     volume = {346},
     number = {17-18},
     year = {2008},
     doi = {10.1016/j.crma.2008.06.014},
     language = {en},
}

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Bo Tan; Zhi-Xiong Wen; Yiping Zhang. On the triplex substitution – Rauzy fractal. Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 925-930. doi : 10.1016/j.crma.2008.06.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.06.014/

Bibliographie
Cité par

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

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

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

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

Cité par Sources :

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

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