### Semi-groupes à générations bornées

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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 $\left|A\right|=2$ 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.

We associate with a word on a finite alphabet an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal of . Then when 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.

Fine and Wilf's 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.

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

Considering each occurrence of a word in a recurrent infinite word, we define the set of return words of to be the set of all distinct words beginning with an occurrence of and ending exactly just before the next occurrence of 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...

In this paper, we survey the rich theory of infinite which generalize to any finite alphabet, in a rather resembling way, the well-known family of on two letters. After recalling definitions and basic properties, we consider 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...

Sia S un semigruppo finitamente generato. Se per ogni coppia x,y di elementi di S si ha che xy oppure yx è un idempotente allora S è finito.

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