Abstract not found

this paper is to present a new proof, with some improvements, of a theorem by Mignosi [21] cited below. Let x be an infinite word, and let F (x) be the sets of its factors (subwords). For w 2 F (x), the index of w in x is the greatest integer d such that w

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.

Sturmian words are in nite words over a two-letter alphabet that admit a great number of equivalent de nitions. Most of them have been given in the past ten years. Among several extensions of Sturmian words to larger alphabets, the Arnoux-Rauzy words appear to share many of the properties of Sturmian words. In this survey, combinatorial properties of these two families are considered and compared.

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. Key words: combinatorics on words; lexicographic order; episturmian word; Sturmian word; Arnoux-Rauzy sequence; balanced word; skew word; episkew word 2000 MSC: 68R15 1

This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz (2003), who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the k-bonacci word, a generalization of the Fibonacci word to a k-letter alphabet (k≥2).

We describe some combinatorial properties of an intriguing class of infinite words connected with the one defined by Kolakoski, defined as the fixed point of the run-length encoding #. It is based on a bijection on the free monoid over # = {1, 2}, that shows some surprising mixing properties. All words contain the same finite number of square factors, and consequently they are cube-free. This suggests that they have the same complexity as confirmed by extensive computations. We further investigate the occurrences of palindromic subwords. Finally we show that there exist smooth words obtained as fixed points of substitutions (realized by transducers) as in the case of K. 1

Abstract not found

It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly (n + 1) factors of length n. One possible generalization of Sturmian sequences is the set of infinite sequences over a k-letter alphabet, k ≥ 3, which are closed under reversal and have at most one right special factor for each length. This is the set of episturmian sequences. These are not necessarily balanced over a k-letter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel’s conjecture for the class of episturmian sequences. This conjecture was first introduced in number theory and has remained unsolved for more than 30 years. It states that for a fixed k> 2, there is only one way to cover Z by k Beatty sequences. The problem can be translated to combinatorics on words: for a k-letter alphabet, there exists only one balanced sequence up to letter permutation that has different letter frequencies. 1

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and “episkew words ” that generalize the skew words of Morse and Hedlund. Keywords: combinatorics on words; episturmian words; Arnoux-Rauzy sequences; Sturmian words; episturmian morphisms. MSC (2000): 68R15. 1