Cohomology of One-dimensional Mixed Substitution Tiling Spaces

We compute the Cech cohomology with integer coefficients of one-dimensional tiling spaces arising from not just one, but several different substitutions, all acting on the same set of tiles. These calculations involve the introduction of a universal version of the Anderson-Putnam complex. We show that, under a certain condition on the substitutions, the projective limit of this universal Anderson-Putnam complex is isomorphic to the tiling space, and we introduce a simplified universal Anderson-Putnam complex that can be used to compute Cech cohomology. We then use this simplified complex to place bounds on the rank of the first cohomology group of a one-dimensional substitution tiling space in terms of the number of tiles.Comment: 26 pages, 4 figure

Matching rules for quasicrystals : the composition-decomposition method

A general method is presented which proves that an appropriately chosen set of matching rules for a quasiperiodic tiling enforces quasiperiodicity. This method, which is based on self-similarity, is formulated in general terms to make it applicable to many different situations. The method is then illustrated with two examples, one of which is a new set of matching rules for a dodecagonal tiling

Some mathematical problems arising in the study of quasicrystals

After briefly reviewing the projection method, some properties of quasicrystal structures obtained by projection are discussed. In particular, the connection between the Fourier spectra and the local isomorphism class of a quasicrystal is worked out. It is argued that quasicrystals have no residual entropy at T = 0. Finally, certain self-similarity properties of quasicrystals are discussed

Dynamics and topology of the Hat family of tilings

The recently discovered Hat tiling admits a 4-dimensional family of shape deformations, including the 1-parameter family already known to yield alternate monotiles. The continuous hulls resulting from these tilings are all topologically conjugate dynamical systems, and hence have the same dynamics and topology. We construct and analyze a self-similar element of this family called the CAP tiling, and we use it to derive properties of the entire family. The CAP tiling has pure-point dynamical spectrum, which we compute explicitly, and comes from a natural cut-and-project scheme with 2-dimensional Euclidean internal space. All other members of the Hat family, in particular the original version constructed from 30-60-90 right triangles, are obtained via small modifications of the projection from this cut-and-project scheme.Comment: 30 pages, 10 figures, 3 tables; slightly expanded and improved version with additional explanations and result

Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings

The two main techniques for the generation of quasipcriodic tilings.,de Bruijn's grid method and the projection formalism, are generalised. A very broad class or quasi periodic tilings is obtained in this way. The two generalised methods are Shown to be equivalent. The standard calculation of Fourier spectra is extended to the whole general class of tilings. Various upplications are discussed