Aperiodic points in $\mathbb Z^2$-subshifts

TL;DR

This paper investigates the structure of aperiodic points in two-dimensional subshifts, providing characterizations of their existence, computational complexity, and implications for higher-dimensional shifts.

Contribution

It establishes a link between large minimal periods and the existence of aperiodic points, and characterizes the structure of subshifts without aperiodic points.

Findings

Presence of arbitrarily large minimal periods implies existence of aperiodic points.

Deciding if a finite type subshift contains an aperiodic point is computationally characterized.

Subshifts without aperiodic points are nearly conjugate to one-dimensional subshifts.

Abstract

We consider the structure of aperiodic points in $\mathbb Z^2$-subshifts, and in particular the positions at which they fail to be periodic. We prove that if a $\mathbb Z^2$-subshift contains points whose smallest period is arbitrarily large, then it contains an aperiodic point. This lets us characterise the computational difficulty of deciding if an $\mathbb Z^2$-subshift of finite type contains an aperiodic point. Another consequence is that $\mathbb Z^2$-subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some $\mathbb Z$-subshift. Finally, we use this result to characterize sets of possible slopes of periodicity for $\mathbb Z^3$-subshifts of finite type.