The Thue-Morse shift, Baumslag-Solitar group, and biminimality

TL;DR

This paper explores the Thue-Morse dynamical system, demonstrating its minimality but not biminimality, and connects it to automata theory and group actions, revealing new links between symbolic dynamics and topological groups.

Contribution

It provides a symbolic encoding of the Thue-Morse system using automata and establishes its minimality without biminimality, linking it to group actions and limit space presentations.

Findings

Thue-Morse system is minimal but not biminimal.

Automata encoding links to Nekrashevych's limit spaces.

Connections made between symbolic dynamics and topological groups.

Abstract

Call a group action on a topological space \emph{biminimal} if for any points $x,y\in X$ there exists a group element taking $x$ arbitrarily close to $y$ and whose inverse takes $y$ arbitrarily close to $x$. A symbolic encoding of the Thue-Morse dynamical system is given, in terms of $\omega$-automata. It is used to prove that the Thue-Morse dynamical system is minimal but not biminimal. The $\omega$-automata also establish a link between Nekrashevych's presentation of limit spaces and solenoids with a construction described by Vershik and Solomyak.