Cobham's theorem for substitutions

Fabien Durand (1)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée

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 $\alpha$ 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 ultimately periodic.

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]; Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]; Systèmes dynamiques [math.DS]

Fabien Durand. Cobham's theorem for substitutions. Journal of the European Mathematical Society, 2011. ⟨hal-00527622⟩