Critical factorisation in square-free words

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 eta(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 eta(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.