Bifix codes and Sturmian words

We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set F of words (F -maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let F be a Sturmian set of words. Our results express the fact that an F -maximal bifix code of degree d behaves just as the set of words of F of length d. An F -maximal bifix code of degree d in a Sturmian set of words has d+1 elements. This generalizes the fact that a Sturmian set contains d + 1 words of length d. Moreover, given an infinite word x, if there is a finite maximal bifix code X of degree d such that x has at most d factors of length d in X, then x is ultimately periodic. We also prove that any F -maximal bifix code of degree d is the basis of a subgroup of index d of the free group on the alphabet.