Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent result (of the second author) that an infinite word is Sturmian if and only if each of its factors has exactly two return words in it. Then, considering episturmian infinite words, which are a natural generalization of Sturmian words, we study the position of the occurrences of any factor in such infinite words and we determinate the return words. At last, we apply these results in order to get a kind of balance property of episturmian words and to calculate the recurrence function of these words.

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Nous présentons une nouvelle caractérisation des mots de Sturm basée sur les mots de retour. Si l'on considère chaque occurrence d'un mot w dans un mot infini récurrent, on définit l'ensemble des mots de retour de w comme l'ensemble de tous les mots distincts commençant par une occurrence de w et finissant exactement avant l'occurrence suivante de w. Le résultat principal montre qu'un mot est sturmien si et seulement si pour chaque mot w non vide apparaissant dans la suite, la cardinalité de l'ensemble des mots de retour de w estégaleà deux.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 w 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 w appearing in the infinite word, the cardinality of the set of return words over w is equal to two.