Polygonal ${\mathbb{Z}}^2$-subshifts

Let $\mathcal{P}\subset {\mathbb{Z}}^2$ be a convex polygon with each vertex in it labeled by an element from a finite set and such that the labeling of each vertex $v\in \mathcal{P}$ is uniquely determined by the labeling of all other points in the polygon. We introduce a class of ${\mathbb{Z}}^2$-shift systems, the {\em polygonal shifts}, determined by such a polygon: these are shift systems such that the restriction of any $x\in X$ to some polygon $\mathcal{P}$ has this property. These polygonal systems are related to various well studied classes of shift systems, including subshifts of finite type and algebraic shifts, but include many other systems. We give necessary conditions for a ${\mathbb{Z}}^2$-system $X$ to be polygonal, in terms of the nonexpansive subspaces of $X$, and under further conditions can give a complete characterization for such systems.