A short proof of a theorem of Cobham on substitutions

TL;DR

This paper provides a simple proof that infinite systems generated by constant length substitutions that are topologically conjugate must have substitution lengths as powers of the same integer, linking dynamical systems and theoretical computer science.

Contribution

It offers a straightforward proof of a special case of Cobham's theorem relating substitution lengths and topological conjugacy in dynamical systems.

Findings

Infinite conjugate systems have substitution lengths as powers of the same integer

The proof simplifies understanding of Cobham's theorem in dynamical systems

Connects substitution lengths with algebraic properties in topological conjugacy

Abstract

This paper is concerned with the lengths of constant length substitutions that generate topologically conjugate systems. We show that if the systems are infinite, then these lengths must be powers of the same integer. This result is a dynamical formulation of a special case of a 1969 theoretical computer science result of Alan Cobham. Our proof is rather simple.