On the structure of Thue-Morse subwords, with an application to dynamical systems

TL;DR

This paper provides a detailed analysis of Thue-Morse subwords, demonstrating the existence of infinitely many primitive substitutions with eigenvalue 2 that produce systems topologically conjugate to the Thue-Morse dynamical system.

Contribution

It introduces a novel characterization of Thue-Morse subwords and constructs infinitely many primitive substitutions with specific eigenvalues related to the system.

Findings

Existence of infinitely many injective primitive substitutions with eigenvalue 2

These substitutions generate systems topologically conjugate to Thue-Morse

Deepened understanding of the structure of Thue-Morse subwords

Abstract

We give an in depth analysis of the subwords of the Thue-Morse sequence. This allows us to prove that there are infinitely many injective primitive substitutions with Perron-Frobenius eigenvalue 2 that generate a symbolic dynamical system topologically conjugate to the Thue-Morse dynamical system.