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

@article{JTNB_1996__8_2_377_0,
     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 = {http://archive.numdam.org/item/JTNB_1996__8_2_377_0/}
}

TY  - JOUR
AU  - Dekking, F. M.
AU  - Wen, Z.-Y.
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  - http://archive.numdam.org/item/JTNB_1996__8_2_377_0/
LA  - en
ID  - JTNB_1996__8_2_377_0
ER  -

%0 Journal Article
%A Dekking, F. M.
%A Wen, Z.-Y.
%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 http://archive.numdam.org/item/JTNB_1996__8_2_377_0/
%G en
%F JTNB_1996__8_2_377_0

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. http://archive.numdam.org/item/JTNB_1996__8_2_377_0/

Bibliographie
Cité par

[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