Smallest Period in Low Complexity Subshifts
Abhishek Khetan · arXiv (Cornell University) · 2021
Let $X$ be a low complexity subshift of $\{0, 1\}^{\mathbb Z^2}$. We show that the directions of periodicity in a minimal weakly periodic decomposition of any weakly periodic point can be detected from the annihilator ideal associated to the subshift. We show that the order of a weakly periodic point is same as the number of components in any minimal decomposition into $1$-periodic elements. We then give an upper bound on the order of a low complexity configuration in terms of the support of any of its annihilators. In the special case of tilings this gives an upper bound on the order of any tiling in terms of a geometric quantity associated to the tile. Lastly we prove that any low complexity subshift of $\{0, 1\}^{\mathbb Z^2}$ whose every point is $1$-periodic has a uniformly bounded smallest period.