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

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.