ON SELF-SIMILARITIES OF CUT-AND-PROJECT SETS

Abstract

Among the commonly used mathematical models of quasicrystals are Delone sets constructed using a cut-and-project scheme, the so-called cut-and-project sets. A cut-and-project scheme (<em>L</em>,π₁, π₂) is given by a lattice <em>L</em> in R^s and projections π₁, π₂ to suitable subspaces V₁, V₂. In this paper we derive several statements describing the connection between self-similarity transformations of the lattice <em>L</em> and transformations of its projections π₁(<em>L</em>), π₂(<em>L</em>). For a self-similarity of a set Σ we take any linear mapping A such that AΣ ⊂ Σ, which generalizes the notion of self-similarity usually restricted to scaled rotations. We describe a method of construction of cut-and-project scheme such that π₁(<em>L</em>) ⊂ R² is invariant under an isometry of order 5. We describe all linear self-similarities of the scheme thus constructed and show that they form an 8-dimensional associative algebra over the ring Z. We perform an example of a cut-and-project set with linear self-similarity which is not a scaled rotation.