A Step toward an MLD Classification of Selfsimilar Quasilattices

Abstract

The point inflation rule (PIR) proposed in a previous paper as a method of obtaining a kind of selfsimilar quasilattices (SSQLs) is extended so that it is applicable to all kinds of SSQLs. The result will be an important step toward a complete MLD classification of SSQLs. The PIR is manifested by an affine autonomous set map (AASM) Ψ characterized by a pair {S ,σ } of a star S and an expansive affine transformation σ; S is a subset of the module L supporting the SSQL and σ is an automorphism of L .I t represents a local rule combining an SSQL Q and its inflation σQ; S specifies the range affected by the local rule. The conjugate map Ψ ⊥ operating on the internal space is another AASM characterized by the conjugate pair {S ⊥ ,σ ⊥ }; σ ⊥ is a contractive affine transformation. The window of Q is a fixed set of Ψ ⊥ and has usually a fractal boundary. The double-star AASM in which two disjoint substars of S play different roles is of particular importance. We produce by maps of this type a lot of new SSQLs with the octagonal, decagonal, and dodecagonal point symmetries. SSQLs of nonBravais type and tilings of tiles with fractal boundaries are included in the formalism. Subject Index: 013