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

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.

Publisher URL: http://arxiv.org/abs/1805.08829

DOI: arXiv:1805.08829v1

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