Undecidability of dynamical properties of SFTs and sofic subshifts on $\mathbb{Z}^2$ and other groups

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.