Critical factorisation in square-free words

07/20/2021

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by   Tero Harju, et al.

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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.