Characterization Of Sturmian Words Research Articles

Palindromic prefixes and episturmian words

Let w be an infinite word on an alphabet A . We denote by ( n i ) i ⩾ 1 the increasing sequence (assumed to be infinite) of all lengths of palindromic prefixes of w. In this text, we give an explicit construction of all words w such that n i + 1 ⩽ 2 n i + 1 for all i, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity lim sup n i + 1 / n i , and prove that it is minimal (among all nonperiodic words) for the Fibonacci word.

Occurrences of palindromes in characteristic Sturmian words

This paper is concerned with palindromes occurring in characteristic Sturmian words c α of slope α , where α ∈ ( 0 , 1 ) is an irrational. As c α is a uniformly recurrent infinite word, any (palindromic) factor of c α occurs infinitely many times in c α with bounded gaps. Our aim is to completely describe where palindromes occur in c α . In particular, given any palindromic factor u of c α , we shall establish a decomposition of c α with respect to the occurrences of u. Such a decomposition shows precisely where u occurs in c α , and this is directly related to the continued fraction expansion of α .

Words derivated from Sturmian words

A return word of a factor of a Sturmian word starts at an occurrence of that factor and ends exactly before its next occurrence. Derivated words encode the unique decomposition of a word in terms of return words. Vuillon has proved that each factor of a Sturmian word has exactly two return words. We determine these two return words, as well as their first occurrence, for the prefixes of characteristic Sturmian words. We then characterize words derivated from a characteristic Sturmian word and give their precise form. Finally, we apply our results to obtain a new proof of the characterization of characteristic Sturmian words which are fixed points of morphisms.

Sturmian words and a criterium by Michaux–Villemaire

Michaux and Villemaire's proof of Cobham's theorem relies on the characterization of ultimately periodic words by means of the behaviour of certain repetitions in the word. Namely, they consider the length of the smallest shift between repetitions of a given length and the first position at which that smallest shift is observed. In this paper we study those properties for characteristic Sturmian words. In particular we answer a question posed by Michaux and Villemaire in that context.

Various Properties of Sturmian Words

This overview paper is devoted to Sturmian words. The first part summarizes different characterizations of Sturmian words. Besides the well known theorem of Hedlund and Morse it also includes recent results on the characterization of Sturmian words using return words or palindromes. The second part deals with substitution invariant Sturmian words, where we present our recent results. We generalize one-sided Sturmian words using the cut-and-project scheme and give a full characterization of substitution invariant Sturmian words.

Conjugates of characteristic Sturmian words generated by morphisms

This article is concerned with characteristic Sturmian words of slope α and 1− α (denoted by c α and c 1− α resp.), where α∈(0,1) is an irrational number such that α=[0;1+d 1, d 2,…,d n ] with d n ≥ d 1≥1. It is known that both c α and c 1− α are fixed points of non-trivial (standard) morphisms σ and σ ̂ , respectively, if and only if α has a continued fraction expansion as above. Accordingly, such words c α and c 1− α are generated by the respective morphisms σ and σ ̂ . For the particular case when α=[0;2, r ̄ ] (r≥1) , we give a decomposition of each conjugate of c α (and hence c 1− α ) into generalized adjoining singular words, by considering conjugates of powers of the standard morphism σ by which it is generated. This extends a recent result of Levé and Séébold on conjugates of the infinite Fibonacci word.

A Characterization of Sturmian Words by Return Words

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 est égale à 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.

Lyndon factorization of sturmian words

We express any general characteristic sturmian word as a unique infinite non-increasing product of Lyndon words. Using this identity, we give a new ω-division for characteristic sturmian words. We also give a short proof of a result by Berstel and de Luca (Sturmian words, Lyndon words and trees, Theoret. Comput. Sci., 178 (1997) 171–203.); more precisely, we show that the set of factors of sturmian words that qualify as Lyndon words is the set of primitive Christoffel words.

Lyndon words and singular factors of Sturmian words

Two different factorizations of the Fibonacci infinite word were given independently in Wen and Wen (1994) and Melançon (1996). In a certain sense, these factorizations reveal a selfsimilarity property of the Fibonacci word. We first describe the intimate links between these two factorizations. We then propose a generalization to characteristic sturmian words.