Abstract
A position p in a word w is critical if the minimal local period at p is equal to the global period of w. According to the Critical Factorisation Theorem all words of length at least two have a critical point. We study the number η(w) of critical points of square-free ternary words w, i.e., words over a three letter alphabet. We show that the sufficiently long square-free words w satisfy η(w) ≤|w|− 5 where |w| denotes the length of w. Moreover, the bound |w|− 5 is reached by infinitely many words. On the other hand, every square-free word w has at least |w|∕4 critical points, and there is a sequence of these words closing to this bound.
Highlights
The Critical Factorisation Theorem [2, 4] is one of the gems in combinatorics on words. It states that each word w with |w| ≥ 2 has a critical point, i.e., a position where the local period ∂(w, p) is equal to the global period ∂(w) of the word
It was shown there that each binary word w of length |w| ≥ 5 and period ∂(w) > |w|/2 has less than |w|/2 critical points
We shall study the number of critical points in ternary square-free words
Summary
The Critical Factorisation Theorem [2, 4] is one of the gems in combinatorics on words. Say w ∈ {0, 1}∗, it was shown in [5] that there are words having only one critical point; e.g., the Fibonacci words of length at least five are such. It was shown there that each binary word w of length |w| ≥ 5 and period ∂(w) > |w|/2 has less than |w|/2 critical points. Each sufficiently long square-free word w can have at most |w| − 5 critical points, and the bound |w| − 5 is obtained by infinitely many square-free w. We prove that a square-free word w has at least |w|/4 critical points, and that there is a sequence of square-free words closing to this bound
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