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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 $y$ andwhose 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 isminimal but not biminimal.  The $\omega$-automata also establish a link between Nekrashevych'spresentation of limit spaces and solenoids with a construction described byVershik and Solomyak.

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