Low-Complexity Tilings of the Plane

Abstract

A two-dimensional configuration is a coloring of the infinite grid Z 2 with finitely many colors. For a finite subset D of Z 2 , the D-patterns of a configuration are the colored patterns of shape D that appear in the configuration. The number of distinct D-patterns of a configuration is a natural measure of its complexity. A configuration is considered having low complexity with respect to shape D if the number of distinct D-patterns is at most |D|, the size of the shape. This extended abstract is a short review of an algebraic method to study periodicity of such low complexity configurations.

Expansivity and periodicity in algebraic subshifts

“…We have discussed a method to infer strong periodicity of a multidimensional configuration from its annihilators or periodizers. The method generalizes the two-dimensional technique used in [5,6,4] to arbitrary dimensions d > 2. The new method is in fact based on a more general condition on the annihilators or periodizers that implies expansivity of a multidimensional subshift in a given direction.…”

“…We start by defining the necessary terminology and concepts. This part is included for the convenience of the reader although it greatly repeats what is written, for example, in [6]. For a vector t ∈ Z d , the translation τ t shifts a configuration c so that the cell t is moved to the cell 0, that is, τ t (c) u = c u+t for all u ∈ Z d .…”

“…Let us finish with some remarks concerning the two-dimensional case d = 2. In this case our tool to infer strong periodicity of a configuration is essentially proved in [5,6] using the structure of the annihilator and periodizer ideals. Non-monomial S-fibers for onedimensional linear subspaces S ⊆ R 2 are called line polynomials as they have at least two monomials and all monomials are along the same line.…”

“…Non-monomial S-fibers for onedimensional linear subspaces S ⊆ R 2 are called line polynomials as they have at least two monomials and all monomials are along the same line. For any two-dimensional configuration c the periodizer ideal Per(c) is known to be a principal ideal φ 1 φ 2 • • • φ m generated by a product of line polynomials φ i [5,6]. If c has a periodizer f that has no line polynomial factors in any direction then from f ∈ φ 1 φ 2 • • • φ m we conclude that m = 0 so that Per(c) = 1 , implying that c is strongly periodic.…”

“…Configurations with annihilators have global rigidity, although they are not necessarily periodic. In the two-dimensional case, periodicity in all directions is known to be enforced if the annihilator has no line polynomial factors, that is, an annihilating polynomial does not have a non-monomial factor whose monomials are on a single line [5,6,4]. In this paper we present a similar condition that works in all dimensions d. More generally, we provide a condition on the annihilator that enforces expansivity: this is a directional determinism property studied in multidimensional symbolic dynamics.…”