Cobham's theorem for substitutions

The seminal theorem of Cobham has given rise during the last 40 years to alot of works around non-standard numeration systems and has been extended tomany contexts. In this paper, as a result of fifteen years of improvements, weobtain a complete and general version for the so-called substitutive sequences.Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers.Then, a sequence $x\in A^\mathbb{N}$, where $A$ is a finite alphabet, is both$\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimatelyperiodic.