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 $\eta (w)\le \mathrm{\mid }w\mathrm{\mid }-5$ where $\mathrm{\mid }w\mathrm{\mid }$ denotes the length of $w$. Moreover, the bound $\mathrm{\mid }w\mathrm{\mid }-5$ is reached by infinitely many words. On the other hand, every square-free word $w$ has at least $\mathrm{\mid }w\mathrm{\mid }\mathrm{/}4$ critical points, and there is a sequence of these words closing to this bound.