Durand, Fabien A theorem of Cobham for nonprimitive substitutions. (English) Zbl 1014.11016 Acta Arith. 104, No. 3, 225-241 (2002). 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. Reviewer: Jean-Paul Allouche (Orsay) Cited in 12 Documents MSC: 11B85 Automata sequences 68R15 Combinatorics on words Keywords:recognizable sets of integers; Cobham theorem; independent numeration bases; substitutions; substitution sequences Citations:Zbl 0179.02501; Zbl 0895.68081 PDFBibTeX XMLCite \textit{F. Durand}, Acta Arith. 104, No. 3, 225--241 (2002; Zbl 1014.11016) Full Text: DOI