On a Rice theorem for dynamical properties of SFTs on groups

Abstract

Let G be a group with undecidable domino problem, such as \({\mathbb {Z}}^2\). We prove that all nontrivial dynamical properties for sofic G-subshifts are undecidable, that this is not true for G-SFTs, and an undecidability result for dynamical properties of G-SFTs similar to the Adian–Rabin theorem. Furthermore, we prove that every computable real-valued dynamical invariant for G-SFTs that is monotone by disjoint unions and products is constant.