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Palindromic complexity of infinite words associatedwith non-simple Parry numbers

Published online by Cambridge University Press:  12 March 2008

L'ubomíra Balková
Affiliation:
Doppler Institute for Mathematical Physics and Applied Mathematics & Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; l.balkova@centrum.cz masakova@km1.fjfi.cvut.cz
Zuzana Masáková
Affiliation:
Doppler Institute for Mathematical Physics and Applied Mathematics & Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; l.balkova@centrum.cz masakova@km1.fjfi.cvut.cz
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Abstract

We study the palindromic complexity of infinite words uβ, the fixed points of the substitution over a binary alphabet, φ(0) = 0a1, φ(1) = 0b1, with a - 1 ≥ b ≥ 1, which are canonically associated with quadratic non-simple Parry numbers β.

Type
Research Article
Copyright
© EDP Sciences, 2008

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References

Ambrož, P., Frougny, Ch., Masáková, Z. and Pelantová, E., Palindromic complexity of infinite words associated with simple Parry numbers. Annales de l'Institut Fourier 56 (2006) 21312160. CrossRef
Baláži, P., Masáková, Z. and Pelantová, E., Factor versus palindromic complexity of uniformly recurrent infinite words. Theor. Comp. Sci. 380 (2007) 266275. CrossRef
L'. Balková, Complexity for infinite words associated with quadratic non-simple Parry numbers. J. Geom. Sym. Phys. 7 (2006) 111.
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) 307328. CrossRef
L'. Balková, E. Pelantová and W. Steiner, Sequences with constant number of return words. Monatshefte fur Mathematik, to appear.
J. Bernat, Étude sur le β-développement et applications. Mémoire de D.E.A., Université de la Méditerrannée Aix-Marseille (2002).
Burdík, Č., Frougny, Ch., Gazeau, J.P. and Krejcar, R., Beta-integers as natural counting systems for quasicrystals. J. Phys. A 31 (1998) 64496472. CrossRef
Damanik, D. and Zamboni, L.Q., Combinatorial properties of Arnoux-Rauzy subshifts and applications to Schrödinger operators. Rev. Math. Phys. 15 (2003) 745763. CrossRef
Damanik, D. and Zare, D., Palindrome complexity bounds for primitive substitution sequences. Discrete Math. 222 (2000) 259267. CrossRef
Fabre, S., Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219236. CrossRef
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.
Ch. Frougny, Z. Masáková, E. Pelantová, Infinite special branches in words associated with beta-expansions. Discrete Math. Theor. Comput. Sci. 9 (2007) 125144.
Hof, A., Knill, O. and Simon, B., Singular continuous spectrum for palindromic Schrödinger operators. Commun. Math. Phys. 174 (1995) 149159. CrossRef
Lagarias, J., Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21 (1999) 161191. CrossRef
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.
Parry, W., On the beta-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401416. CrossRef
Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477493. CrossRef
Schmidt, K., On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269278. CrossRef
Shechtman, D., Blech, I., Gratias, D. and Cahn, J., Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984) 19511954. CrossRef
W.P. Thurston, Groups, tilings, and finite state automata. Geometry supercomputer project research report GCG1, University of Minnesota (1989).
Turek, O., Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO-Theor. Inf. Appl. 41 (2007) 123135. CrossRef