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

## TL;DR

This paper introduces polygonal ${
m Z}^2$-shift systems, a new class of shift systems characterized by convex polygons with uniquely determined vertex labelings, linking them to known classes like subshifts of finite type.

## Contribution

The paper defines polygonal ${
m Z}^2$-shift systems, explores their properties, and provides necessary conditions and characterizations based on nonexpansive subspaces.

## Key findings

- Polygonal shifts generalize several known shift classes.
- Necessary conditions relate to nonexpansive subspaces.
- Complete characterization possible under certain conditions.

## Abstract

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.