# Density of Critical Factorizations

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 36, Issue: 3, page 315-327
- ISSN: 0988-3754

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topHarju, Tero, and Nowotka, Dirk. "Density of Critical Factorizations." RAIRO - Theoretical Informatics and Applications 36.3 (2010): 315-327. <http://eudml.org/doc/92704>.

@article{Harju2010,

abstract = {
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.
},

author = {Harju, Tero, Nowotka, Dirk},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Combinatorics on words; repetitions;
Critical Factorization Theorem; density of critical
factorizations; Fibonacci words; Thue–Morse words.; infinite sequences of words; critical factorization},

language = {eng},

month = {3},

number = {3},

pages = {315-327},

publisher = {EDP Sciences},

title = {Density of Critical Factorizations},

url = {http://eudml.org/doc/92704},

volume = {36},

year = {2010},

}

TY - JOUR

AU - Harju, Tero

AU - Nowotka, Dirk

TI - Density of Critical Factorizations

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 36

IS - 3

SP - 315

EP - 327

AB -
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.

LA - eng

KW - Combinatorics on words; repetitions;
Critical Factorization Theorem; density of critical
factorizations; Fibonacci words; Thue–Morse words.; infinite sequences of words; critical factorization

UR - http://eudml.org/doc/92704

ER -

## References

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- J.-P. Duval, Périodes et répétitions des mots de monoïde libre. Theoret. Comput. Sci.9 (1979) 17-26.
- M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, Massachusetts, Encyclopedia of Math. 17 (1983).
- M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, Cambridge, United Kingdom (2002).
- M. Morse, Recurrent geodesics on a surface of negative curvature. Trans. Amer. Math. Soc.22 (1921) 84-100.
- A. Thue, Über unendliche Zeichenreihen. Det Kongelige Norske Videnskabersselskabs Skrifter, I Mat.-nat. Kl. Christiania 7 (1906) 1-22.
- A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Det Kongelige Norske Videnskabersselskabs Skrifter, I Mat.-nat. Kl. Christiania 1 (1912) 1-67.

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