A characterization of sturmian words by return words

We present a new characterization of Sturmian words using return words. Considering each occurrence of a word w in a recurrent word, we define the set of return words over tv to be the set of all distinct words beginning with an occurrence of w and ending exactly before the next occurrence of w in the infinite word. It is shown that an infinite word is a Sturmian word if and only if for each non-empty word to appearing in the infinite word, the cardinality of the set of return words over w is equal to two. (C) 2001 Academic Press.