Boundedness of oriented walks generated by substitutions
Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 377-386.

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

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

DOI: 10.5802/jtnb.175
Keywords: substitutions, self-similarity, walks
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F. M. Dekking; Z.-Y. Wen. Boundedness of oriented walks generated by substitutions. Journal de Théorie des Nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 377-386. doi : 10.5802/jtnb.175. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.175/

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