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Sur des points fixes de morphismes d’un monoïde libre. (French) Zbl 0691.68065

Logicians define natural numbers as follows: \(A_ 0=\emptyset =\{\) \(\}\), \(A_ 1=\{\emptyset \}=\{\{\) \(\}\), \(\{\) \(\{\) \(\}\) \(\}\) \(\}\), etc. Replacing symbols \(\{\) and \(\}\) by a and b respectively, one can write: \(A_ 0=ab\), \(A_ 1=aabb\), \(A_ 2=aabaabbb,...,A_{n+1}=aA_ 0A_ 1...A_ nb,... \). It is shwn that the “limit” of \(A_ n\) (when n tends to \(+\infty)\) is an infinite word A which is a fixed point of the endomorphism \(a\to aab\), \(b\to b\) of the monoid \(\{a,b\}^*\). The structure of A is investigated and the methods developed for that purpose are applied to the study of several other infinite sequences. It is proved that there exists a formal series, algebraic over the field of rational fractions modulo 3, which is neither rational nor quadratic and whose sequence of partial quotients is generated by a 3-automaton (this is a partial answer to a conjecture of Mendès-France).
Reviewer: S.M.Goberstein

MSC:

68Q70 Algebraic theory of languages and automata
20M35 Semigroups in automata theory, linguistics, etc.
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References:

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