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Abstract. In this paper we investigate powers of prefixes of Sturmian sequences. We give an explicit formula for ice(ω), the initial critical exponent of a Sturmian sequence ω, defined as the supremum of all real numbers p>0 for which there exist arbitrary long prefixes of ω of the form u p, in terms of its S-adic representation. This formula is based on Ostrowski’s numeration system. Furthermore we characterize those irrational slopes α of which there exists a Sturmian sequence ω beginning in only finitely many powers of 2 + ε, that is for which ice(ω) =2. In the process we recover the known results for the index (or critical exponent) of a Sturmian sequence. We also focus on the Fibonacci Sturmian shift and prove that the set of Sturmian sequences with ice strictly smaller than its everywhere value has

We study powers of prefixes of Sturmian words. 1.

Abstract. Several results on continued fractions expansions are direct on indirect consequences of the mirror formula. We survey occurrences of this formula for Sturmian real numbers, for (simultaneous) Diophantine approximation, and for formal power series. 1.

The combinatorial properties of the Fibonacci infinite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity, quasicrystals etc. In this note, we introduce the singular words of the Fibonacci infinite word and discuss their properties. We establish two decompositions of the Fibonacci word in singular words and their consequences. By using these results, we discuss the local isomorphism of the Fibonacci word and the overlap properties of the factors. Moreover, we also give new proofs for the results on special words and the power of the factors. The combinatorial properties of the Fibonacci infinite word are of great interest in some aspects of mathematics and physics, such as number theory, fractal geometry, formal language, computational complexity, quasicrystals etc. See [1, 3, 8, 9, 11]. Moreover, the properties of the subwords of the Fibonacci infinite word have been studied extensiv...

We show that Dejean’s conjecture holds for n ≥ 27. Repetitions in words have been studied since the beginning of the previous century [13, 14]. Recently, there has been much interest in repetitions with fractional exponent [1, 3, 5, 6, 7, 9]. For rational 1 < r ≤ 2, a fractional r-power is a non-empty word w = xx ′ such that x ′ is the prefix of x of length (r − 1)|x|. For example, 010 is a 3/2-power. A basic problem is that of identifying the repetitive threshold for each alphabet size n> 1: What is the infimum of r such that an infinite sequence on n letters exists, not containing any factor of exponent greater than r? The infimum is called the repetitive threshold of an n-letter alphabet, denoted by RT(n). Dejean’s conjecture [5] is that ⎨ 7/4, n = 3 RT(n) = 7/5, n = 4 n/(n − 1) n ̸ = 3, 4 Thue, Dejean and Pansiot, respectively [14, 5, 12] established the values RT(2), RT(3), RT(4). Moulin-Ollagnier [11] verified Dejean’s conjecture for 5 ≤ n ≤ 11, and Mohammad-Noori and Currie [10] proved the conjecture for 12 ≤ n ≤ 14.

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved in [7] and [9]. 1

We consider several open problems of Karhumaki, Mignosi, and Plandowski, cf. [KMP], concerning the expressibility of languages and relations as solutions of word equations. We show first that the (scattered) subword relation is not expressible. Then, we consider the set of k-power free finite words and solve it negativelly for all nontrivial integer values of k. Finally, we consider the Fibonacci finite words. We do not solve the problem of the expressibility of the set of these words but prove that it cannot be given a negative answer (as believed) using the tools in [KMP]. 1

Je remercie Jacques Désarménien pour ses conseils scientifiques et pratiques dans la direction du travail de ma thèse ces dernières années. Il a su maintenir vivant mon rêve de soutenir une thèse d’état. Grazie, Jacques! Je remercie Jean Berstel. En lisant ses articles sur la combinatoire des mots, toujours clairs et intéressants, j’ai beaucoup appris. Grazie Jean! Merci aussi à Christian Choffrut. Christian connaissait parfaitement les importants travaux de Justin sur les semigroupes répétitifs. Il m’a toujours prodigué ses précieux conseils. Sans lui, ma thèse de troisième cycle n’aurait pas pu voir le jour. Grazie, Christian! Je remercie aussi Maxime Crochemore, que je connais depuis de longues années, pour les nombreuses et fructueuses discussions que nous avons eues.

each α> 2 there is an infinite binary word with