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Enumeration of factors in the Thue-Morse word. (English) Zbl 0683.20045

The infinite word of Thue-Morse M is the limit \(\lim_{n\to \infty}\phi^ n(a)\), where \(\phi\) is the endomorphism of the free monoid \(\{a,b\}^*\) defined by \(\phi (a)=ab\), \(\phi (b)=ba\). The author proves that if M is written \(w_ 1w_ 2\), then either \(w_ 1\) has a square suffix, or \(w_ 2\) has a square prefix. A byproduct of his methods allows him to deduce a result of Pansiot, which characterizes the square factors of M. Finally, he gives some results on the function \(P(m)=number\) of factors of length m of M, in particular \[ \liminf_{m\to \infty}P(m)/(m-1)=3\quad and\quad \limsup P(m)/(m- 1)=10/3. \]
Reviewer: Ch.Reutenauer

MSC:

20M05 Free semigroups, generators and relations, word problems
05A15 Exact enumeration problems, generating functions
20M35 Semigroups in automata theory, linguistics, etc.
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