Squirals and beyond: substitution tilings with singular continuous spectrum

The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on${ \mathbb{Z} }^{d} $.

C*-Algebras Associated with Mauldin–Williams Graphs

A Mauldin–Williams graph is a generalization of an iterated function system by a directed graph. Its invariant set K plays the role of the self-similar set. We associate a C*-algebra (K) with a Mauldin–Williams graph and the invariant set K, laying emphasis on the singular points. We assume that the underlying graph G has no sinks and no sources. If satisfies the open set condition in K, and G is irreducible and is not a cyclic permutation, then the associated C*-algebra (K) is simple and purely infinite. We calculate the K-groups for some examples including the inflation rule of the Penrose tilings.

THE TRIANGLE PATTERN — A NEW QUASIPERIODIC TILING WITH FIVEFOLD SYMMETRY

We introduce a quasiperiodic tiling with fivefold symmetry that is built from two types of triangles, an acute and an obtuse one. An easy to computerize construction algorithm based on the dualization scheme is presented that creates the pattern not pointwise but tile by tile directly. We present the vertex statistics, the inflation rule, and the Fourier analysis for selected decorations with pointlike atoms. A connection to the planar Penrose pattern and to the 3D icosahedral quasilattice is briefly discussed.