Cobham's theorem for substitutions

TL;DR

This paper extends Cobham's theorem to substitutive sequences, showing that sequences that are substitutive with respect to two multiplicatively independent Perron numbers are ultimately periodic.

Contribution

It provides a complete and general version of Cobham's theorem for substitutive sequences, based on fifteen years of research improvements.

Findings

Sequences substitutive for two multiplicatively independent Perron numbers are ultimately periodic.

The result generalizes previous partial theorems in the context of non-standard numeration systems.

The theorem applies to a broad class of substitutive sequences, unifying various prior results.

Abstract

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.