A word is called closed if it has a prefix which is also its suffix and there is no internal occurrences of this prefix in the word. In this paper we study words that are rich in closed factors, i.e., which contain the maximal possible number of distinct closed factors. As the main result, we show that for finite words the asymptotics of the maximal number of distinct closed factors in a word of length $n$ is $\frac{n^2}{6}$. For infinite words, we show there exist words such that each their factor of length $n$ contains a quadratic number of distinct closed factors, with uniformly bounded constant; we call such words infinite closed-rich. We provide several necessary and some sufficient conditions for a word to be infinite closed rich. For example, we show that all linearly recurrent words are closed-rich. We provide a characterization of rich words among Sturmian words. Certain examples we provide involve non-constructive methods.
翻译：如果一个单词有一个前缀, 也就是它的后缀, 并且字中没有这个前缀的内部发生。 在本文中, 我们研究的单词中含有大量封闭因素, 即包含最大可能数量的独特封闭因素。 作为主要结果, 我们显示对于限定词来说, 一个长度单词中不同封闭因素的最大数目的无症状是 $\ frac{ n2\\\ 6}$。 对于无限的单词, 我们显示的单词有这样的单词, 其长度 $n$ 的每个因数都包含一系列不同的封闭因素, 并且具有一致的固定不变性; 我们称之为无限封闭性。 我们为一个单词提供了一些必要和足够的条件, 以使一个单词成为无限封闭的丰富。 例如, 我们显示所有线性重复的单词都是封闭性的。 我们给出了Sturmian 单词中丰富单词的特征。 我们提供的某些例子涉及非构造方法 。
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