A Characterization of Sturmian Words by Return Words

Abstract

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.

Itineraries Induced by Exchange of Two Intervals

“…, T −n (α)} for some n ∈ N, then the set of Iitineraries R(ρ) for ρ ∈ I consists of only two words. This reformulates the result of Vuillon [4] about the existence of exactly two return words to a fixed factor of a Sturmian word.…”

Sequences with constant number of return words

“…Using return words, Vuillon [13] found a new equivalent definition of Sturmian words. He showed that an infinite word u over a biliteral alphabet is Sturmian if and only if any factor of u has exactly two return words.…”

Recent Results on Extensions of Sturmian Words

“…This is a characteristic property. This was proved by Vuillon [45]. A simplified proof appears in [35].…”