Substitutive systems and a finitary version of Cobham's theorem

TL;DR

This paper investigates substitutive systems generated by nonprimitive substitutions, proving transitive subsystems are also substitutive, and characterizes common factors of automatic sequences over multiplicatively independent bases, extending Cobham's theorem.

Contribution

It provides a new characterization of substitutive systems and generalizes Cobham's theorem to a broader class of automatic sequences.

Findings

Transitive subsystems of substitutive systems are themselves substitutive.

Characterization of common factors of automatic sequences over multiplicatively independent bases.

Extension of Cobham's theorem to nonprimitive substitutions.

Abstract

We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear as common factors of two automatic sequences defined over multiplicatively independent bases. This generalises the famous theorem of Cobham.