Classification of sofic projective subdynamics of multidimensional shifts of finite type

Abstract:

Motivated by Hochman’s notion of subdynamics of a $\mathbb {Z}^d$ subshift (2009), we define and examine the projective subdynamics of $\mathbb {Z}^d$ shifts of finite type (SFTs) where we restrict not only the action but also the phase space. We show that any $\mathbb {Z}$ sofic shift of positive entropy is the projective subdynamics of a $\mathbb {Z}^2$ ($\mathbb {Z}^d$) SFT, and that there is a simple condition characterizing the class of zero-entropy $\mathbb {Z}$ sofic shifts which are not the projective subdynamics of any $\mathbb {Z}^2$ SFT. We define notions of stable and unstable subdynamics in analogy with the notions of stable and unstable limit sets in cellular automata theory, and discuss how our results fit into this framework. One-dimensional strictly sofic shifts of positive entropy admit both a stable and an unstable realization, whereas $\mathbb {Z}$ SFTs only allow for stable realizations and a particular class of zero-entropy proper $\mathbb {Z}$ sofics only allows for an unstable realization. Finally, we prove that the union of finitely many $\mathbb {Z}^k$ subshifts, all of which are realizable in $\mathbb {Z}^d$ SFTs, is again realizable when it contains at least two periodic points, that the projective subdynamics of $\mathbb {Z}^2$ SFTs with the uniform filling property (UFP) are always stable, thus sofic, and we exhibit a class of non-sofic $\mathbb {Z}$ subshifts which are not the projective subdynamics of any $\mathbb {Z}^d$ SFT.