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Let $A$ be a finite set, a subshift of $A^{\mathbb{Z}^2}$ is a closed subset $X\subseteq A^{\mathbb{Z}^2}$ (with respect to the product topology, $A$ as a finite set is of course considered to be discrete) that is also closed under the canonical shift action of $\mathbb{Z}^2$. $X$ is of finite type (SFT) if it is defined by finitely many forbidden patterns, i.e. there is a finite subset $F\subseteq \mathbb{Z}^2$ and a finite subset $\mathcal{F}\subseteq A^F$ such that $$X:=\{x\in A^{\mathbb{Z}^2}\colon \forall g\in\mathbb{Z}^2\;\forall p\in\mathcal{F}\; (g\cdot x\upharpoonright F\neq p)\}.$$

It is known that there are SFTs for $\mathbb{Z}^2$ that are aperiodic, i.e. the stabilizer of each point of such SFT is trivial. I wonder if such aperiodic SFTS may be minimal as dynamical systems, that is, having no non-trivial proper subshifts, or there is some reason that prevents that from happening.

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A $\mathbb{Z}^2$-SFT may or may not be minimal. The aperiodic Robinson tileset defines an SFT that is not minimal (see e.g. https://arxiv.org/abs/1203.1387). On the other hand, there exist minimal SFTs, which can be very complicated (there are better references for just the existence of minimal SFTs, but see e.g. https://arxiv.org/abs/1705.01876).

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