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Density of Critical Factorizations

Published online by Cambridge University Press:  15 December 2002

Tero Harju  and
Dirk Nowotka
Tero Harju
Affiliation:
Turku Centre for Computer Science, TUCS, and Department of Mathematics, University of Turku; [email protected]. [email protected].
Dirk Nowotka
Affiliation:
Turku Centre for Computer Science, TUCS, and Department of Mathematics, University of Turku; [email protected]. [email protected].
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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.

Keywords

Combinatorics on wordsrepetitionsCritical Factorization Theoremdensity of critical factorizationsFibonacci wordsThue–Morse words.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 36 , Issue 3 , July 2002 , pp. 315 - 327
Copyright
© EDP Sciences, 2002

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References

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