A generalization of Cobham's Theorem

TL;DR

This paper generalizes Cobham's Theorem by showing that if a sequence is the image of a fixed point of two primitive substitutions, then their dominant eigenvalues are multiplicatively dependent.

Contribution

It introduces a new generalization of Cobham's Theorem relating primitive substitutions and eigenvalues of their matrices.

Findings

Sequences from two primitive substitutions have multiplicatively dependent eigenvalues.

The generalization applies to non-periodic sequences as images of fixed points.

Provides a new perspective on the structure of sequences related to primitive substitutions.

Abstract

If a non-periodic sequence $X$ is the image by a morphism of a fixed point of both a primitive substitution $\sigma$ and a primitive substitution $\tau$, then the dominant eigenvalues of the matrices of $\sigma$ and of $\tau$ are multiplicatively dependent. This is the way we propose to generalize Cobham's Theorem.