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 RECOGNIZABILITY FOR SEQUENCES OF MORPHISMS by Valérie Berthé, Wolfgang Steiner, Jörg M. Thuswaldner and Reem 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.

Strictly speaking my conjecture stays open for erasing substitutions, but I should emphasize that the authors solve precisely the issue I was interested in; I just didn't myself assume non-erasing, because I assumed something much stronger (at least morally), and I do not know any dynamical behaviors exhibited by erasing substitutions but no non-erasing ones (whereas the growing condition I used in my paper seems quite restrictive). (On the other hand I also have no proof or reason to believe erasing substitutions do not generate new behaviors.)

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 \(\mathbb{N}\).

Last update: 8 May 2018