Aperiodic points in $\mathbb Z^2$-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.
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Anaël Grandjean, Benjamin Hellouin de Menibus, Pascal Vanier. Aperiodic points in $\mathbb Z^2$-subshifts. ICALP 2018, Jul 2018, Prague, Czech Republic. ⟨10.4230/LIPIcs.ICALP.2018.496⟩. ⟨hal-01722008v2⟩

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