Boundedness of oriented walks generated by substitutions

Dekking, F. M.; Wen, Z.-Y.

Dekking, F. M. ; Wen, Z.-Y.

Journal de théorie des nombres de Bordeaux, Tome 8 (1996) no. 2, pp. 377-386.

Résumé
Abstract

Soit $x = x_{0} x_{1} \dots$ un point fixe de la substitution sur l’alphabet $\{a, b\},$ et soit $U_{a} = (\begin{matrix} - 1 & - 1 \\ 0 & 1 \end{matrix})$ et $U_{b} = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ . On donne une classification complète des substitutions $σ : {\{a, b\}}^{☆}$ selon que la suite de matrices ${(U_{x_{0}} U_{x_{1}} \dots U_{x_{n}})}_{n = 0}^{\infty}$ est bornée ou non. Cela correspond au fait que les chemins orientés engendrés par les substitutions sont bornés ou non.

Let $x = x_{0} x_{1} \dots$ be a fixed point of a substitution on the alphabet $\{a, b\},$ and let $U_{a} = (\begin{matrix} - 1 & - 1 \\ 0 & 1 \end{matrix})$ and $U_{b} = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$ . We give a complete classification of the substitutions $σ : {\{a, b\}}^{☆}$ according to whether the sequence of matrices ${(U_{x_{0}} U_{x_{1}} \dots U_{x_{n}})}_{n = 0}^{\infty}$ is bounded or unbounded. This corresponds to the boundedness or unboundedness of the oriented walks generated by the substitutions.

MR Zbl

Mots-clés : substitutions, self-similarity, walks

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     author = {Dekking, F. M. and Wen, Z.-Y.},
     title = {Boundedness of oriented walks generated by substitutions},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {377--386},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
     number = {2},
     year = {1996},
     mrnumber = {1438476},
     zbl = {0869.11020},
     language = {en},
     url = {https://numdam.org/item/JTNB_1996__8_2_377_0/}
}

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Dekking, F. M.; Wen, Z.-Y. Boundedness of oriented walks generated by substitutions. Journal de théorie des nombres de Bordeaux, Tome 8 (1996) no. 2, pp. 377-386. https://numdam.org/item/JTNB_1996__8_2_377_0/

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