# Cobham's theorem for substitutions

### Fabien Durand

Université de Picardie Jules Verne, Amiens, France

## Abstract

The seminal theorem of Cobham has given rise during the last 40 years to a lot of works about 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.

## Cite this article

Fabien Durand, Cobham's theorem for substitutions. J. Eur. Math. Soc. 13 (2011), no. 6, pp. 1799–1814

DOI 10.4171/JEMS/294