Palindromic complexity of infinite words associated with non-simple Parry numbers

Balková, L'ubomíra; Masáková, Zuzana

doi:10.1051/ita:2008005

Balková, L'ubomíra ; Masáková, Zuzana

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 145-163.

Résumé

We study the palindromic complexity of infinite words $u_{β}$ , the fixed points of the substitution over a binary alphabet, $ϕ (0) = 0^{a} 1$ , $ϕ (1) = 0^{b} 1$ , with $a - 1 \geq b \geq 1$ , which are canonically associated with quadratic non-simple Parry numbers $β$ .

MR Zbl

DOI : 10.1051/ita:2008005

Classification : 68R15, 11A63
Mots clés : palindromes, beta-expansions, infinite words

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     author = {Balkov\'a, L'ubom{\'\i}ra and Mas\'akov\'a, Zuzana},
     title = {Palindromic complexity of infinite words associated with non-simple {Parry} numbers},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {145--163},
     publisher = {EDP-Sciences},
     volume = {43},
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     year = {2009},
     doi = {10.1051/ita:2008005},
     mrnumber = {2483448},
     zbl = {1156.68043},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2008005/}
}

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Balková, L'ubomíra; Masáková, Zuzana. Palindromic complexity of infinite words associated with non-simple Parry numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 145-163. doi : 10.1051/ita:2008005. http://www.numdam.org/articles/10.1051/ita:2008005/

Bibliographie
Cité par

[1] P. Ambrož, Ch. Frougny, Z. Masáková and E. Pelantová, Palindromic complexity of infinite words associated with simple Parry numbers. Annales de l'Institut Fourier 56 (2006) 2131-2160. | Numdam | MR | Zbl

[2] P. Baláži, Z. Masáková and E. Pelantová, Factor versus palindromic complexity of uniformly recurrent infinite words. Theor. Comp. Sci. 380 (2007) 266-275. | MR | Zbl

[3] L’. Balková, Complexity for infinite words associated with quadratic non-simple Parry numbers. J. Geom. Sym. Phys. 7 (2006) 1-11. | MR | Zbl

[4] L’. Balková, E. Pelantová and O. Turek, Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers. RAIRO-Theor. Inf. Appl. 41 (2007) 307-328. | Numdam | MR | Zbl

[5] L’. Balková, E. Pelantová and W. Steiner, Sequences with constant number of return words. Monatshefte fur Mathematik, to appear.

[6] J. Bernat, Étude sur le $β$ -développement et applications. Mémoire de D.E.A., Université de la Méditerrannée Aix-Marseille (2002).

[7] Č. Burdík, Ch. Frougny, J.P. Gazeau and R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. Phys. A 31 (1998) 6449-6472. | MR | Zbl

[8] D. Damanik and L.Q. Zamboni, Combinatorial properties of Arnoux-Rauzy subshifts and applications to Schrödinger operators. Rev. Math. Phys. 15 (2003) 745-763. | MR | Zbl

[9] D. Damanik and D. Zare, Palindrome complexity bounds for primitive substitution sequences. Discrete Math. 222 (2000) 259-267. | MR | Zbl

[10] S. Fabre, Substitutions et $β$ -systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | MR | Zbl

[11] Ch. Frougny, Z. Masáková and E. Pelantová, Complexity of infinite words associated with beta-expansions. RAIRO-Theor. Inf. Appl. 38 (2004), 163-185; Corrigendum. RAIRO-Theor. Inf. Appl. 38 (2004) 269-271. | Numdam | MR | Zbl

[12] Ch. Frougny, Z. Masáková and E. Pelantová, Infinite special branches in words associated with beta-expansions. Discrete Math. Theor. Comput. Sci. 9 (2007) 125-144. | MR | Zbl

[13] A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys. 174 (1995) 149-159. | MR | Zbl

[14] J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21 (1999) 161-191. | MR | Zbl

[15] Y. Meyer, Quasicrystals, Diophantine approximation, and algebraic numbers, in Beyond Quasicrystals, edited by F. Axel, D. Gratias. EDP Sciences, Les Ulis; Springer, Berlin (1995) 6-13. | MR | Zbl

[16] W. Parry, On the beta-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401-416. | MR | Zbl

[17] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477-493. | MR | Zbl

[18] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269-278. | MR | Zbl

[19] D. Shechtman, I. Blech, D. Gratias and J. Cahn, Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984) 1951-1954.

[20] W.P. Thurston, Groups, tilings, and finite state automata. Geometry supercomputer project research report GCG1, University of Minnesota (1989).

[21] O. Turek, Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO-Theor. Inf. Appl. 41 (2007) 123-135. | Numdam | MR | Zbl

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