Density of critical factorizations
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 36 (2002) no. 3, pp. 315-327.

We investigate the density of critical factorizations of infinite sequences of words. The density of critical factorizations of a word is the ratio between the number of positions that permit a critical factorization, and the number of all positions of a word. We give a short proof of the Critical Factorization Theorem and show that the maximal number of noncritical positions of a word between two critical ones is less than the period of that word. Therefore, we consider only words of index one, that is words where the shortest period is larger than one half of their total length, in this paper. On one hand, we consider words with the lowest possible number of critical points and show, as an example, that every Fibonacci word longer than five has exactly one critical factorization and every palindrome has at least two critical factorizations. On the other hand, sequences of words with a high density of critical points are considered. We show how to construct an infinite sequence of words in four letters where every point in every word is critical. We construct an infinite sequence of words in three letters with densities of critical points approaching one, using square-free words, and an infinite sequence of words in two letters with densities of critical points approaching one half, using Thue-Morse words. It is shown that these bounds are optimal.

DOI: 10.1051/ita:2002016
Classification: 68R15
Keywords: combinatorics on words, repetitions, critical factorization theorem, density of critical factorizations, Fibonacci words, Thue-Morse words
@article{ITA_2002__36_3_315_0,
     author = {Harju, Tero and Nowotka, Dirk},
     title = {Density of critical factorizations},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {315--327},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {3},
     year = {2002},
     doi = {10.1051/ita:2002016},
     zbl = {1013.68154},
     mrnumber = {1958246},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2002016/}
}
TY  - JOUR
AU  - Harju, Tero
AU  - Nowotka, Dirk
TI  - Density of critical factorizations
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2002
DA  - 2002///
SP  - 315
EP  - 327
VL  - 36
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita:2002016/
UR  - https://zbmath.org/?q=an%3A1013.68154
UR  - https://www.ams.org/mathscinet-getitem?mr=1958246
UR  - https://doi.org/10.1051/ita:2002016
DO  - 10.1051/ita:2002016
LA  - en
ID  - ITA_2002__36_3_315_0
ER  - 
%0 Journal Article
%A Harju, Tero
%A Nowotka, Dirk
%T Density of critical factorizations
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2002
%P 315-327
%V 36
%N 3
%I EDP-Sciences
%U https://doi.org/10.1051/ita:2002016
%R 10.1051/ita:2002016
%G en
%F ITA_2002__36_3_315_0
Harju, Tero; Nowotka, Dirk. Density of critical factorizations. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 36 (2002) no. 3, pp. 315-327. doi : 10.1051/ita:2002016. http://www.numdam.org/articles/10.1051/ita:2002016/

[1] J. Berstel, Axel Thue's work on repetitions in words, edited by P. Leroux and Ch. Reutenauer, Séries formelles et combinatoire algébrique. Université du Québec à Montréal, Publications du LaCIM 11 (1992) 65-80.

[2] J. Berstel, Axel Thue's papers on repetitions in words: A translation. Université du Québec à Montréal, Publications du LaCIM 20 (1995).

[3] Y. Césari and M. Vincent, Une caractérisation des mots périodiques. C. R. Hebdo. Séances Acad. Sci. 286(A) (1978) 1175-1177. | MR | Zbl

[4] Ch. Choffrut and J. Karhumäki, Combinatorics of words, edited by G. Rozenberg and A. Salomaa. Springer-Verlag, Berlin, Handb. Formal Languages 1 (1997) 329-438. | MR

[5] M. Crochemore and D. Perrin, Two-way string-matching. J. ACM 38 (1991) 651-675. | MR | Zbl

[6] A. De Luca, A combinatorial property of the Fibonacci words. Inform. Process. Lett. 12 (1981) 193-195. | MR | Zbl

[7] J.-P. Duval, Périodes et répétitions des mots de monoïde libre. Theoret. Comput. Sci. 9 (1979) 17-26. | MR | Zbl

[8] M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, Massachusetts, Encyclopedia of Math. 17 (1983). | MR | Zbl

[9] M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, Cambridge, United Kingdom (2002). | MR | Zbl

[10] M. Morse, Recurrent geodesics on a surface of negative curvature. Trans. Amer. Math. Soc. 22 (1921) 84-100. | JFM | MR

[11] A. Thue, Über unendliche Zeichenreihen. Det Kongelige Norske Videnskabersselskabs Skrifter, I Mat.-nat. Kl. Christiania 7 (1906) 1-22. | JFM

[12] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Det Kongelige Norske Videnskabersselskabs Skrifter, I Mat.-nat. Kl. Christiania 1 (1912) 1-67. | JFM

Cited by Sources: