Randomness and Topological Invariants in Pentagonal Tiling Spaces

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.

CONFIGURATIONAL ENTROPY FOR STONE-INFLATION HEXAGONAL AND OCTAGONAL PATTERNS

In a recent paper several species of octagonal patterns have been introduced with the help of a construction which allows us to derive them by means of inflation rules. Non-deterministic patterns can be generated by composition of the inflation rules. In this paper we show how a similar construction produces patterns with hexagonal symmetry. The non-deterministic rhombus–triangle tilings are obtained by local rearrangements of tiles which are included in the inflation rules. This property allows to compute the configurational entropy.

A Construction of Inflation Rules for Pisot Octagonal Tilings

A construction for the generation of eight-fold symmetry planar patterns is introduced. The basic building blocks are four triangle prototiles with six edge lengths. Subpatterns with three prototile shapes and two edge lengths can also be derived.