# Среда 24.10. Jarkko Kari (University of Turku): "Periodicity in Algebraic Subshifts"

A two-dimensional configuration over the alphabet B={0,1} is a binary coloring Z^2 -> B of the infinite grid. For a finite subset D of Z^2, the D-patterns of a configuration are the patterns of shape D that appear in the configuration. The algebraic subshift defined by shape D is the set of the configurations whose D-patterns all contain an even number of 1s. For example, the Ledrappier subshift consists of those configurations where each cell is colored by the modulo 2 sum of the colors on its north and north-east neighbors. We say a configuration has low complexity if it has at most |C| different C-patterns, for some finite C. Every low complexity configuration is in some algebraic subshift so to understand the periodicity properties of general low complexity configurations it is enough to consider elements of algebraic subshifts. It turns out that in the Ledrappier subshift all low complexity configurations are periodic, and the same is true for any algebraic subshift whose defining shape D has line factors in at most one direction.