On the Number of Closed Factors in a Word

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special.We investigate the structure of closed factors of words. We show that a word of length n contains at least n + 1 distinct closed factors, and characterize those words having exactly n + 1 closed factors. Furthermore, we show that a word of length n can contain Θ(n²) many distinct closed factors.