Abstract
We analyze substitution tiling spaces with fivefold symmetry. In the substitution process, the introduction of randomness can be done by means of two methods which may be combined: composition of inflation rules for a given prototile set and tile rearrangements. The configurational entropy of the random substitution process is computed in the case of prototile subdivision followed by tile rearrangement. When aperiodic tilings are studied from the point of view of dynamical systems, rather than treating a single one, a collection of them is considered. Tiling spaces are defined for deterministic substitutions, which can be seen as the set of tilings that locally look like translates of a given tiling. Čech cohomology groups are the simplest topological invariants of such spaces. The cohomologies of two deterministic pentagonal tiling spaces are studied.
Highlights
Aperiodic tilings of the plane appeared in the literature in the works of Wang 1 and Penrose 2
We study the cohomology of Penrose tiling spaces ΩP along the lines of 8, 18 but with different inflation rules
Apart from the vertex configurations and relative orientations, an essential difference between the spaces of pentagonal tilings ΩΞ, ΩΞ− and the very well-known space of Penrose tilings ΩP is the cohomology, in spite of the fact that the first inflation step in the tiling growth seems to be very similar
Summary
Aperiodic tilings of the plane appeared in the literature in the works of Wang 1 and Penrose 2. A property of quasicrystal structures is the appearance of sharp peaks in their diffraction patterns and recent results in this direction use methods familiar from statistical mechanics and from the long-range aperiodic order of tilings 3. Discrete Dynamics in Nature and Society the atoms in a material modeled with a quasicrystal tiling are distributed in such a way that they determine a quasiperiodic potential. Pentagonal, octagonal, decagonal, and dodecagonal quasicrystalline materials have been found in experiments, and tilings with the corresponding symmetries in the diffraction patterns are candidates of their structural models. We first study the derivation of both deterministic and random tilings and we discuss the cohomology groups of some tilings generated by an inflation process.
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