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

Abstract

We consider the structure of aperiodic points in Z 2 -subshifts, and in particular the positions at which they fail to be periodic. We prove that if a 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 Z 2 -subshift of finite type contains an aperiodic point. Another consequence is that Z 2 -subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some Z-subshift. Finally, we use this result to characterize sets of possible slopes of periodicity for Z 3 -subshifts of finite type. ACM Subject ClassificationTheory of computation → Models of computation Keywords and phrases Subshifts of finite type, Wang tiles, periodicity, aperiodicity, computability, tilings Digital Object Identifier 10.4230/LIPIcs.ICALP.2018.496 Related Version hal-01722008v1Acknowledgements The authors wish to thank anonymous reviewers for many helpful remarks and improvements.A subshift on Z d is a set of colorings of Z d by a finite set of colors avoiding some family of forbidden patterns. When this family is finite, the subshift is called a subshift of finite type (SFT). In dimension 2, SFTs are equivalent to sets of tilings by Wang tiles: Wang tiles are unit squares with colored borders that cannot be rotated and may be placed next to each other only if the borders match.Wang tiles were introduced by Wang in order to study the decidability of some fragments of logic [18,19]. He thus introduced the Domino Problem: given a set of Wang tiles, do they tile the plane? (in other words, is the corresponding subshift nonempty?) Wang first conjectured that whenever a tileset tiles the plane, it can do so in a periodic manner, which would have implied the decidability of the Domino Problem.