In [Bérthé-Steiner-Thuswaldner-Yassawi], Proposition 5.15 states that for every non-erasing substitution there are only finitely many languages for points. This is the same thing as quasiminimality by an easy argument (see e.g. Lemma 11 and Proposition 6 in my paper).

Here's their theorem in my language:

Let \(\tau : \Sigma \to \Sigma^+\) be any substitution. Let \(X_{\tau} \subset \Sigma^{\mathbb{Z}}\) be the subshift consisting of the points \(x \) whose finite subwords all appear in some \(\tau^n(a), n \in \mathbb{N}, a \in \Sigma\). Then \(X_{\tau}\) is a quasiminimal subshift.

Theorem 7.5.1 in [Allouche-Shallit] (also [Cobham]) shows that a morphic image of the fixed point of a (possibly erasing) substitution is also the symbol-to-symbol substitutive image of a fixed point of a non-erasing substitution. It is not difficult to obtain the general result from this and the above theorem, as images of quasiminimal subshifts under non-erasing morphisms are quasiminimal. A similar idea was also suggested by Wolfgang Steiner in private communication.

Let \(\tau : \Sigma \to \Sigma^*\) be any substitution. Let \(X_{\tau} \subset \Sigma^{\mathbb{Z}}\) be the subshift consisting of the points \(x \) whose finite subwords all appear in some \(\tau^n(a), n \in \mathbb{N}, a \in \Sigma\). Then \(X_{\tau}\) is a quasiminimal subshift.

Errata:

- The definition I give of the subshift of a substitution in my paper has an unfortunate typo: In Definition 10, obviously \(j\) and \(k\) should range over \(\mathbb{Z}\) rather than \(\mathbb{N}\).
- Marie-Pierre Béal, Dominique Perrin and Antonio Restivo pointed out to me that the proof of Lemma 3 is wrong. Specifically, the proof of the "In particular" claim (that is, the last sentence) is wrong. Somehow I missed the subtleties here even though I am clearly very aware of the same difficulties when discussing quasiminimality. The only excuse I can think of is
~~incompetence~~that this proof was taken from my thesis draft, and was written for primitive substitutions (for which this proof is fine), and I didn't see where the primitivity is needed so I erased it and forgot about it. I believe I have a working proof sketch of this lemma, but I haven't gotten to writing it down in the half a year that I've known about the error, so at this point I'll just add this note without a correction, hopefully I get to this in finite time. If you are interested, bug me about it and I'll have to get on it!

[Bérthé-Steiner-Thuswaldner-Yassawi] Berthé, Valérie and Steiner, Wolfgang and Thuswaldner, Jörg M. and Yassawi, Reem. (2018) "Recognizability for sequences of morphisms," Ergodic Theory and Dynamical Systems. Cambridge University Press, pp. 1–36. doi: 10.1017/etds.2017.144.

[Cobham] Cobham, Alan. "On the Hartmanis-Stearns problem for a class of tag machines." IEEE Conference Record of 1968 Ninth Annual Symposium on Switching and Automata Theory. IEEE, 1968.

Last update: 1 Mar 2022