I've been looking at several papers which allude to a relation between SFTs. Namely, given an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ with allowed patches $\mathcal{F}$, we can associate a set of domino\Wang tiles $T\_{\mathcal{F}}$ defined by $\mathcal{F}$. The question of whether there is a periodic configuration, an $\omega\in \Omega$ such that $\vert \text{orbit}\_{\mathbb{Z}^2}(\omega)\vert<\infty$, can be determined by studying the domino problem for $T\_{\mathcal{F}}$.
Alternatively, given a set of Wang\domino $T$, we can choose an alphabet $ \mathcal{A}$ and get an SFT $\Omega \subseteq \mathcal{A}^{\mathbb{Z}^2}$ encoding the domino\wang problem of $T$.


Is there a good reference for this sort of relation? I've mainly found this topic discussed shortly in the prelimnaries of several papers. For example,


* [A SELF-SIMILAR APERIODIC SET OF 19 WANG TILES](https://arxiv.org/pdf/1802.03265.pdf) in section 2.
* [On the domino problem of the Baumslag-Solitar groups](https://www.sciencedirect.com/science/article/pii/S030439752100517X) in section 2.2.
* [Multi-Dimensional Symbolic Dynamical Systems](https://link.springer.com/chapter/10.1007/978-1-4613-0165-3_3), in section 3.


I was wondering if there is a good reference dealing with this relation\encoding of Wang tiles and SFTs. All these papers deal with this shortly, but I assumed that there might be a reference dealing with this relation more thoroughly, which I was not able to find.



Reference on relation between SFTs and Wang-tiles
I don't have a good reference at hand but I can explain the procedure. I'll use a quadruple to denote a Wang tile $T = (N,E,S,W)$ referring to the North, East, South and West edges of a tile respectively.


**Easy direction:** Given a set of Wang tiles $\{T\_1, \ldots, T\_k\}$, let $B\_h$ be the set of pairs $(T\_i,T\_j)$ such that $T\_i(E) \neq T\_j(W)$ and let $B\_v$ be the set of pairs $(T\_i,T\_j)$ such that $T\_i(S) \neq T\_j(N)$. Now let $\mathcal{F}\_h$ be the patch of tiles with $T\_j$ to the right of $T\_i$, where $(T\_i, T\_j) \in B\_h$ and let $\mathcal{F}\_v$ be the patch of tiles with $T\_j$ below $T\_i$, where $(T\_i,T\_j) \in B\_v$. We may now take our set of forbidden patches $\mathcal{F}$ to be given by $\mathcal{F}\_h \cup \mathcal{F}\_v$.


**Harder direction:** Let $\mathcal{A}$ be our alphabet. Given a set of forbidden patches $\mathcal{F} = \{P\_i \mid 1 \leq i \leq k\}$, without loss of generality, we may assume that each of the patches is supported on an $n \times n$ square (as otherwise, just take all possible extensions of the patches to an $n \times n$ square such that $n$ is big enough to contain all the patches). Now, let $\mathcal{L}^{n,n}$ be the set of legal $n \times n$ patches. That is, $\mathcal{L}^{n,n} = \mathcal{A}^{n,n}\setminus \mathcal{F}$. Let $P\_N$ be the patch one gets by deleting the northern-most row of $P$. Similarly for $P\_E$, $P\_S$ and $P\_W$ for their respective rows/columns.


For every $P \in \mathcal{L}^{n,n}$, we define a Wang tile $T\_P$ by


$$T\_P(N) = (P\_S, P\_W, P\_N, P\_E).$$


So, the northern edge of $T\_P$ contains all of the information of the patch $P$ except for the southern-most row, etc.


It's not too hard to see that two Wang tiles $T\_P$ and $T\_Q$ can only be placed next to each other if their corresponding North-South or East-West pair are identical, which is the same as saying that the patches $P$ and $Q$ overlap in a legal way.


There is now a simple local rule that takes one from a tiling with the Wang tiles $T\_P$, $P \in \mathcal{L}^{n,n}$ to an element of the original SFT $X\_{\mathcal{F}}$ whereby one just needs to rebuild the original patch $P$ from $T\_P$ (which one can do using the labels of each of the edges) and then outputs the central element of $P$ (or round down if $n$ is even).


