Notes and errata on "Decidability and Universality of Quasiminimal Subshifts"

The main results of my paper are about abstract quasiminimal subshifts under computability restrictions, but there is also a tangent to classical systems, where I prove the following: If a substitution \(\tau\) has the following property: "In every point of the language of the subshift that \(\tau\) generates, in every word of length \(m\) you have a letter \(a\) such that \(|\tau^n(a)|\) tends to infinity." then the subshift \(\tau\) generates is quasiminimal. I conjecture in my paper that in fact every substitution has a quasiminimal associated subshift. In this note, we prove this conjecture.

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.


As for the 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 \(\mathbb{N}\).

References

[Allouche-Shallit] Allouche, Jean-Paul, and Jeffrey Shallit. Automatic sequences: theory, applications, generalizations. Cambridge university press, 2003.
[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: 10 Aug 2018