Communicated byM. Waldschmidt

The automatic sequence is the central concept at the intersection of formal language theory and number theory. It was introduced by Cobham, and has been extensively studied by Christol, Kamae, Mendes France and Rauzy, and other writers. Since the range of automatic sequences is nite, however, their descriptive power is severely limited.

The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.

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.

A word on a finite alphabet A is said to be overlap-free if it contains no factor of the form xuxux, where x is a letter and u a (possibly empty) word. In this paper we study the number un of overlap-free binary words of length n, which is known to be bounded by a polynomial in n. First, we describe a bijection between the set of overlap-free words and a rational language. This yields recurrence relations for un , which allow to compute un in logarithmic time. Then, we prove that the numbers ff = sup f r j n r = O (un) g and fi = inf f r j un = O (n r ) g are distinct, and we give an upper bound for ff and a lower bound for fi. Finally, we compute an asymptotically tight bound to the number of overlap-free words of length less than n. 1 Introduction In general, the problem of evaluating the number un of words of length n in the language U consisting of words on some finite alphabet A with no factors in a certain set F is not easy. If F is finite, it amounts to counting words in...

In recent years, there has been a number of papers about the combinatorial notion of symbolic complexity: this is the function counting the number of factors of length n for a sequence. The complexity is an indication of the degree of randomness of the sequence: a periodic sequence has a bounded complexity, the expansion of a normal number has an exponential complexity. For a given sequence, the complexity function is generally not of easy access, and it is a rich and instructive work to compute it; a survey of this kind of results can be found in [ALL]. We are interested here in further results in the theory of symbolic complexity, somewhat beyond the simple question of computing the complexity of various sequences. These lie mainly in two directions; first, we give a survey of an open question which is still very much in progress, namely: to determine which functions can be the symbolic complexity function of a sequence. Then, we investigate the links between the complexity of a sequence and its associated dynamical system, and insist on the cases where the knowledge of

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

Infinite Lyndon words have been introduced in [1], where the authors proved a factorization theorem for infinite words: any infinite word can be written as a non increasing product of Lyndon words, finite and/or infinite. After giving a new characterization of infinite Lyndon words, we concentrate on three well known infinite words and give their factorization. We conclude by giving an application to !-division of infinite words.

We characterize the squares occurring in infinite overlap-free binary words and construct various α power-free binary words containing infinitely many overlaps. 1

Smooth words are infinite words connected to the Kolakoski sequence. We construct the maximal and the minimal infinite smooth words, with respect to the lexicographical order. The naive algorithm generating these words is improved by using a reduction of the De Bruijn graph of their factors. We also study their Lyndon factorizations. 1