Cobham's theorem for substitutions

The seminal theorem of Cobham has given rise during the last 40 years to a lot of works around non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let α and β be two multiplicatively independent Perron numbers. Then, a sequence x ∈ AN, where A is a finite alphabet, is both α-substitutive and β-substitutive if and only if x is ultimately periodic.