I was wondering whether there are primitive symbolic substitutions over $\mathbb{Z}^d$ and alphabet $\mathcal{A}$ whose associated subshift is equal to an aperiodic SFT. By SFT here I mean a subshift of finite type, as in this link or this thread.
This might be a simple and trivial answer to this, but I was hoping some knowledgeable person will be able to clarify some of my confusion.
My understanding is that in one dimension, this is impossible since we can find a periodic sequence in the substitution subshift. One can see this by finding a cycle in a De-Brujin graph associated to the substitution subshift. I believe that in fact any minimal subshift in one dimension should contain some periodic sequence.
It is my understanding that in $2$ dimensions, the situation is different. I know that there are minimal SFTs which are aperiodic, since we can translate Wang tiles to SFTs in $\mathcal{A}^{\mathbb{Z}^2}$. I was wondering whether there is a constant 'length' substitution, given by $S:\mathcal{A}\to \mathcal{A}^{Q}$ for $Q=\prod_{j=1}^d\{ 0,1,...,m_{j}-1 \}$, such that its susbtitution subshift is an aperiodic SFT? I know of a paper by Sébastien Labbé that gives Wang tiles generated from a substitution, but I am unsure whether the Wang shift is a factor of the substitution subshift. It also does not seem clear to me, whether one can translate this result to a constant length subshift.
I've confused myself over this for some weeks now, and was wondering whether perhaps some knowledgeable source could help clarify this. I would appreciate any clarification on this subject and any new information on this topic.
