## On the periodicity of morphisms on free monoids.(English)Zbl 0608.68065

It is shown that for a given endomorphism h on a given finitely generated free monoid there are only finitely many primitive words w for which $$h(w)=w^ n$$ for some $$n\geq 2$$. Moreover, all such words can be effectively found. Using this result, the D0L periodicity problem is shown to be decidable, that is, it is decidable whether there exist words v and w for a given u such that $$h^{\omega}(u)=vw^{\omega}$$, where $$h^{\omega}(u)$$ is the limit of the sequence $$u,h(u),h^ 2(u),...$$. This latter result has also been solved by J. Pansiot [Decidability of periodicity for infinite words, RAIRO Inf. Théor. 20, 43-46 (1986)] using a different method.

### MSC:

 68Q45 Formal languages and automata 20M35 Semigroups in automata theory, linguistics, etc. 20M05 Free semigroups, generators and relations, word problems 68Q42 Grammars and rewriting systems
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### References:

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