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 $\eta(w) \le |w|-5$ 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.

中文翻译：

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无平方词中的临界因子分解

如果 $p$ 的最小局部周期等于 $w$ 的全局周期，则单词 $w$ 中的位置 $p$ 是关键的。根据临界因子分解定理，所有长度至少为 2 的单词都有一个临界点。我们研究了无平方三元词 $w$ 的临界点数 $\eta(w)$，即三个字母字母表上的词。我们证明足够长的无平方词 $w$ 满足 $\eta(w) \le |w|-5$ 其中 $|w|$ 表示 $w$ 的长度。此外，无限多的词可以达到界限 $|w|-5$。另一方面，每个无方块词 $w$ 至少有 $|w|/4$ 个临界点，并且这些词的序列接近这个界限。