I was reviewing the following statement from a survey by E. Arthur Robinson about tilings in $\mathbb{R}^d$ to better understand geometric tiling rather than tilings over symbols. I consider the notions for substitution in section $4$ of the aforementioned survey. I seem to have stumbled on alleged contradiction which I can't resolve on my own.
I am interested in examples of periodic tilings over substitution space such that all their $k$-patches are legal, for some $k>1$. Namely, a $k$-patch $P\subseteq \mathcal{T}^*$ is legal if $P$ is a sub-patch of $S^n(T)$, for some $T\in \mathcal{T}$ and $n\in \mathbb{N}$. I think that this translates to whether there is a periodic tiling in a finite type tiling space, described in definition $3.3$, with the forbidden patches being the illegal $k$-patches over $\mathcal{T}$.
If I understand Corollary $4.10$, then there can be no such tilings in the case of the Penrose tilings. This follows since substitution on the finite type tiling space, arising from the Penrose tiling, is bijective and invertible. But going by this thread and its reference Nonperiodicity Implies Unique Composition for Self-Similar Translationally Finite Tilings, it seems that any aperiodic substitution tilings will likewise have no such periodic tilings.
This contradicts what I heard about the Octagonal\Ammann–Beenker tiling which have such a periodic tiling, shown by M Duneau in Approximants of quasiperiodic structures generated by the inflation mapping.
Since these two conclusions contradict one another, I was wondering where and what are the faults in this reasoning?
Later edit
I am interested whether one can find a tiling not in the substitution space which satisfies the local matching rule of the substitution.
In the symbolic substitution case over $\mathbb{Z}^d$, with prototiles corresponding to letter in an alphabet $\mathcal{A}$, I think the finite type tiling space capturing the legal $2$-patches, is the SFT by the legal patches with support $\{0,1\}^d$.
There are a lot of examples where this SFT contains periodic tilings\configurations. Some examples include:
- The table tiling substitution space and a configuration of $\omega:= \begin{pmatrix} 1 & 3\\ 3& 1 \end{pmatrix}^\infty$
- The Fibonacci substitution tiling space and a configuration of $\omega:=0^\infty$
- The Thue-Morse substitution tiling space and any periodic configuration $\omega \in \mathcal{A}^\mathbb{Z}$
I was looking for a similar example of an aperiodic geometric substitutions tiling space, with the finite type tiling spaces generated by the legal $2$-patches. Based on the arguments I wrote above, I thought it might be impossible to find examples for geometric substitutions. However, following Ville Salo's comment I think that there may be such an example. Does anyone know of such an example?
