Finite and infinite closed-rich words

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.