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.

Keywords: substitutions, self-similarity, walks
@article{JTNB_1996__8_2_377_0,
author = {F. M. Dekking and Z.-Y. Wen},
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},
zbl = {0869.11020},
mrnumber = {1438476},
language = {en},
url = {https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_377_0/}
}
TY  - JOUR
AU  - F. M. Dekking
AU  - Z.-Y. Wen
TI  - Boundedness of oriented walks generated by substitutions
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1996
SP  - 377
EP  - 386
VL  - 8
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_377_0/
LA  - en
ID  - JTNB_1996__8_2_377_0
ER  -
%0 Journal Article
%A F. M. Dekking
%A Z.-Y. Wen
%T Boundedness of oriented walks generated by substitutions
%J Journal de théorie des nombres de Bordeaux
%D 1996
%P 377-386
%V 8
%N 2
%I Université Bordeaux I
%U https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_377_0/
%G en
%F JTNB_1996__8_2_377_0
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. https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_377_0/

[1] F.M. Dekking, Recurrent sets, Advances in Math. 44 (1982), 78-104. | MR | Zbl

[2] F.M. Dekking, On transience and recurrence of generalized random walks, Z. Wahrsch. verw. Geb. 61 (1982), 459-465. | MR | Zbl

[3] F.M. Dekking, Marches automatiques, J. Théor. Nombres Bordeaux 5 (1993), 93-100. | Numdam | MR | Zbl

[4] F.M. Dekking, Iteration of maps by an automaton, Discrete Math. 126 (1994), 81-86. | MR | Zbl

[5] J.-M. Dumont et A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions, Theor. Comp. Science 65 (1989), 153-169. | MR | Zbl

[6] J.-M. Dumont, Summation formulae for substitutions on a finite alphabet, Number Theory and Physics (Eds: J.-M. Luck, P. Moussa, M. Waldschmidt). Springer Lect. Notes Physics 47 (1990), 185-194. | MR | Zbl

[7] M. Mendès France and J. Shallit, Wirebending and continued fractions, J. Combinatorial Theory Ser. A 50 (1989), 1-23. | MR | Zbl

[8] D. Levine and P.J. Steinhardt, Quasicrystals (I). Definition and structure. Physical Review B, vol. (2) 34, 1986, 596-615. | MR

[9] P.A.B. Pleasants, Quasicrystallography: some interesting new patterns. Banach center publications, vol. 17, 1985, 439-461. | MR | Zbl

[10] Z.-X. Wem and Z.-Y. Wen, Marches sur les arbres homogènes suivant une suite substitutive, J. Théor. Nombres Bordeaux, 4 (1992), 155-186. | Numdam | MR | Zbl