×
Berthé, Valérie

Frequencies of factors of Sturmian sequences. (Fréquences des facteurs des suites sturmiennes.) (French) Zbl 0872.11018

Theor. Comput. Sci. 165, No. 2, 295-309 (1996).
Summary: Dekking a explicité les fréquences des facteurs de la suite de Fibonacci en utilisant le graphe des mots. Nous généralisons ce résultat aux suites sturmiennes en montrant, également par le graphe des mots, que les fréquences des facteurs de même longueur d’une suite sturmienne prennent au plus 3 valeurs. Nous explicitons ces valeurs et donnons, pour chacune d’elles, le nombre de facteurs ayant cette fréquence en fonction du développement en fraction continue de l’angle \(\alpha\) de la suite sturmienne.

Cited in 3 Reviews
Cited in 30 Documents

MSC:

11B85 Automata sequences
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)

Keywords:

Sturmian sequences
PDFBibTeX XMLCite
\textit{V. Berthé}, Theor. Comput. Sci. 165, No. 2, 295--309 (1996; Zbl 0872.11018)
Full Text: DOI

References:

[1] Arnoux, P.; Rauzy, G., Représentation géométrique de suites de complexité \(2n + 1\), Bull. Soc. math. France, 119, 199-215 (1991) · Zbl 0789.28011
[2] Berstel, J., Mots de Fibonacci, Séminaire d’Informatique Théorique, LITP, Universités Paris, 6-7, 57-78 (1980-1981)
[3] Berstel, J.; Séébold, P., Morphismes de Sturm, Bull. Belg. Math. Soc., 1, 175-189 (1994) · Zbl 0803.68095
[4] Berthé, V., Fonctions de Carlitz et automates. Entropies conditionnelles, (Thèse (1994), Univ. Bordeaux I)
[5] Berthé, V., Conditional entropy of some automatic sequences, J. Phys. A: Math. Gen., 27, 7993-8006 (1994) · Zbl 0846.11019
[6] Brown, T. C., Descriptions of the characteristic sequence of an irrational, Can. Math. Bull., 36, 15-21 (1993) · Zbl 0804.11021
[7] Burrows, B. L.; Sulston, K. W., Measures of disorder in non-periodic sequences, J. Phys. A: Math. Gen., 24, 3979-3987 (1991) · Zbl 0751.11015
[8] Coven, E. M.; Hedlund, G. A., Sequences with minimal block growth, Math. Systems Theory, 7, 138-153 (1973) · Zbl 0256.54028
[9] Crisp, D.; Moran, W.; Pollington, A.; Shiue, P., Substitution invariant cutting sequences, J. Théorie Nombres Bordeaux, 5, 123-137 (1993) · Zbl 0786.11041
[10] Dekking, F. M., On the Prouhet-Thue-Morse Measure, Acta Universitatis Carolinae, Mathematica et Physica, 33, 35-40 (1992) · Zbl 0790.11017
[11] A. De Luca et F. Mignosi, Some combinatorial properties of Sturmian words, Theoret. Comput. Sci.; A. De Luca et F. Mignosi, Some combinatorial properties of Sturmian words, Theoret. Comput. Sci. · Zbl 0874.68245
[12] Fine, N. J.; Wilf, H. S., Uniquenes theorems for periodic functions, (Proc. Amer. Math. Soc., 16 (1965)), 109-114 · Zbl 0131.30203
[13] Hardy, G. H.; Wright, E. M., An Introduction to The Theory of Numbers (1979), Oxford Science Publications: Oxford Science Publications Oxford · Zbl 0423.10001
[14] P. Hubert, Communication privée.; P. Hubert, Communication privée.
[15] Mignosi, F., Infinite words with linear subword complexity, Theoret. Comput. Sci., 65, 221-242 (1989) · Zbl 0682.68083
[16] Mignosi, F., On the number of factors of Sturmian words, Theoret. Comput. Sci., 82, 71-84 (1991) · Zbl 0728.68093
[17] Morse, M.; Hedlund, G. A., Symbolic dynamics, Amer. J. Math., 60, 815-866 (1938) · JFM 64.0798.04
[18] Morse, M.; Hedlund, G. A., Symbolic dynamics II: Sturmian trajectories, Amer. J. Math., 62, 1-42 (1940) · JFM 66.0188.03
[19] Rauzy, G., Suites à termes dans un alphabet fini, (Sém. de Théorie des Nombres de Bordeaux (1983)), 25dash01-25dash16 · Zbl 0547.10048
[20] Rauzy, G., Mots infinis en arithmétique, (Nivat, M.; Perrin, D., Automata on Infinite Words. Automata on Infinite Words, Lecture Notes in Computer Science, Vol. 192 (1985), Springer: Springer Berlin), 165-171 · Zbl 0613.10044
[21] Slater, N. B., Gaps and steps for the sequence nθ mod 1, (Proc. Cambridge Phil. Soc., 63 (1967)), 1115-1123 · Zbl 0178.04703
[22] Sós, V. T., On the distribution mod 1 of the sequence nα, Ann. Univ. Sci. Budapest Eötvös Sect. Math., 1, 127-134 (1958) · Zbl 0094.02903
[23] Stolarsky, K. B., Beatty sequences, continued fractions, and certain shift operators, Can. Math. Bull., 19, 473-482 (1976) · Zbl 0359.10028
[24] Świerczkowski, S., On successive settings of an arc on the circumference of a circle, Fundam. Math., 46, 187-189 (1958) · Zbl 0085.27203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.