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A theorem of Cobham for nonprimitive substitutions. (English) Zbl 1014.11016

A theorem of A. Cobham [Math. Syst. Theory 3, 186-192 (1969; Zbl 0179.02501)] asserts that a sequence which is both the pointwise image of a fixed point of a uniform morphism of length \(p\) and the pointwise image of a fixed point of a uniform morphism of length \(q\) must be ultimately periodic if \(p\) and \(q\) are multiplicatively independent. Many papers have been devoted to trying to generalize this result. A general conjecture, formulated by G. Hansel, states that if a sequence is both \(\alpha\)-substitutive and \(\beta\)-substitutive, with \(\alpha\) and \(\beta\) multiplicatively independent, then the sequence must be ultimately periodic. Here “\(\alpha\)-substitutive” means “pointwise image of a fixed point of a morphism whose transition matrix has its dominant eigenvalue equal to \(\alpha\)”.
The author of the paper under review previously proved that the conjecture is true if the morphisms are primitive [see Theory Comput. Syst. 31, 169-185 (1998; Zbl 0895.68081)]. Here some other classes of morphisms are proved to be “good” for the conjecture.

MSC:

11B85 Automata sequences
68R15 Combinatorics on words
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