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

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