Matching Rules for Substitution and Hierarchical Tilings for any Substitution with Finite Local Complexity

Abstract

The Goodman-Strauss theorem states that for ``almost every'' substitution $τ$, the family of substitution tilings is sofic, that is, it can be defined by local matching rules for some decoration of tiles. The conditions on the substitution that guarantee the soficity are quite complicated in the statement of the theorem. In this paper we propose a version of the Goodman-Strauss theorem with very simple conditions on the substitution: the family of substitution tilings must have finite local complexity (FLC), that is, the number of crowns that appear in $τ$-supertiles is finite. Like the original theorem, our theorem provides matching rules for all known substitution tilings. We also prove a similar theorem for the family of \emph{hierarchical} tilings associated with the given substitution. A tiling is called $τ$-hierarchical if it has a composition under $τ$, such that this composition also has a composition, and so on, infinitely many times. Every substitution tiling is hierarchical, but the converse is not always true.