Jacques Justin

Episturmian words and some constructions of de Luca and Rauzy

Abstract In this paper we study infinite episturmian words which are a natural generalization of Sturmian words to an arbitrary alphabet. A characteristic property is: they are closed under reversal and have at most one right special factor of each length. They are first obtained by a construction due to de LUCA which utilizes the palindrome closure. They can also be obtained by the way of extended RAUZY rules.

Episturmian words and episturmian morphisms

Infinite episturmian words are a generalization of Sturmian words which includes the Arnoux-Rauzy sequences. We continue their study and that of episturmian morphisms, begun previously, in relation with the action of the shift operator. Palindromic and periodic factors of these words are described. We consider, in particular, the case where these words are generated by morphisms and introduce then a notion of intercept generalizing that of Sturmian words. Finally, we prove that the frequencies of the factors in a strong sense do exist for all episturmian words.

On a paper by Castelli, Mignosi, Restivo

Fine and Wilfs theorem has recently been extended to words having three periods. Following the method of the authors we extend it to an arbitrary number of periods and deduce from that a characterization of generalized Arnoux-Rauzy sequences or episturmian infinite words.

Episturmian morphisms and a Galois theorem on continued fractions

We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of w . Then when |A|=2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

Fractional powers in Sturmian words

Abstract Given an infinite Sturmian word s , we calculate the function L(m) which gives the length of the longest factor of s having period m . The expression of L(m) makes use of the continued fraction of the irrational α associated with s .

EPISTURMIAN WORDS: SHIFTS, MORPHISMS AND NUMERATION SYSTEMS

Episturmian words, which include the Arnoux-Rauzy sequences, are infinite words on a finite alphabet generalizing the Sturmian words and sharing many of their same properties. This was studied in previous papers. Here we gain a deeper insight into these properties. This leads in particular to consider numerations systems similar to the Ostrowski ones and to give a matrix formula for computing the number of representations of an integer in such a system. We also obtain a complete answer to the question: if an episturmian word is morphic, which shifts of it, if any, also are morphic ?

On a characteristic property of ARNOUX–RAUZY sequences

Here we give a characterization of Arnoux-Rauzy sequences by the way of the lexicographic orderings of their alphabet.

Characterizations of finite and infinite episturmian words via lexicographic orderings

In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a wide sense (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian.

On a natural extension of Jacob's ranks

As shown by van der Waerden in his celebrated theorem, if the set of positive integers is partitioned into finitely many classes, then at least one of these classes contains an arithmetic progression of any given length. It is rather trivial to see that every class may contain no arithmetic progression of infinite length; one can show even more: each class may contain no arithmetic progression of arbitrary length with bounded common difference. In our terminology (see below), the above mentioned results can be rephrased by saying that the “van der Waerden congruence” (i.e., two words of a finitely generated free semigroup are equivalent if and only if they have the same first letter and the same length) is repetitive, but it is not strongly repetitive. On the other hand, a congruence related to the preceding one, in which two words are equivalent if and only if they have the same first letter, is strongly repetitive (Brown’s lemma, [ 11). This result is generalized by Theorem 1 (announced in [6]), which provides a characterization of finite semigroups. Jacob, by using a collection of functions, called “ranks,” obtained a different generalization of Brown’s lemma (see [3]). This notion of “rank” suggests, as a natural extension, our definition of “hypomorphism” which is a function which maps a semigroup into an ordered semigroup. This allows us to prove Theorem 2 (announced also in [6], which contains both Theorem 1 and Jacob’s result as particular cases.

Decimations and sturmian words

Standard Sturmian infinite words have a curious property, discovered by G. Rauzy. If in such a word we delete all occurrences of each letter, except every pth one, then we get the same infinite word. This property and several generalizations are studied here. In the last part a short and self-contained theory of Sturmian words, using only combinatorial arguments, is presented.