We study the algorithmic undecidability of abstract dynamical properties for sofic $\mathbb{Z}^{2}$-subshifts and subshifts of finite type (SFTs) on $\mathbb{Z}^{2}$. Within the class of sofic $\mathbb{Z}^{2}$-subshifts, we prove the undecidability of every nontrivial dynamical property. We show that although this is not the case for $\mathbb{Z}^{2}$-SFTs, it is still possible to establish the undecidability of a large class of dynamical properties. This result is analogous to the Adian-Rabin undecidability theorem for group properties. Besides dynamical properties, we consider dynamical invariants of $\mathbb{Z}^{2}$-SFTs taking values in partially ordered sets. It is well known that the topological entropy of a $\mathbb{Z}^{2}$-SFT can not be effectively computed from an SFT presentation. We prove a generalization of this result to \emph{every} dynamical invariant which is nonincreasing by factor maps, and satisfies a mild additional technical condition. Our results are also valid for $\Z^{d}$, $d\geq2$, and more generally for any group where determining whether a subshift of finite type is empty is undecidable.

翻译：我们研究了 $\mathbb{Z}^2$ 上 sofic 子移位与有限型子移位 (SFT) 的抽象动力性质的算法不可判定性。在 sofic $\mathbb{Z}^2$-子移位类中，我们证明了每一个非平凡动力性质的不可判定性。我们指出，尽管这一结论对 $\mathbb{Z}^2$-SFT 并不成立，但仍可建立一大类动力性质的不可判定性。该结果类似于群性质中的 Adian-Rabin 不可判定性定理。除了动力性质之外，我们还考虑取值为偏序集的 $\mathbb{Z}^2$-SFT 的动力不变量。众所周知，$\mathbb{Z}^2$-SFT 的拓扑熵无法通过 SFT 表示有效计算。我们将此结果推广至所有通过因子映射非增且满足一个温和额外技术条件的动力不变量。我们的结论对 $\mathbb{Z}^d$（$d\geq2$）同样成立，更一般地，对任何判定有限型子移位是否为空不可判定的群均成立。