Randomness and Topological Invariants in Pentagonal Tiling Spaces | Zendy

Juan Garcı́a Escudero | Zendy

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Randomness and Topological Invariants in Pentagonal Tiling Spaces

Author(s) -

Juan Garcı́a Escudero

Publication year - 2011

Publication title -

discrete dynamics in nature and society

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.264

H-Index - 39

eISSN - 1607-887X

pISSN - 1026-0226

DOI - 10.1155/2011/946913

Subject(s) - substitution tiling , aperiodic graph , mathematics , tessellation (computer graphics) , substitution (logic) , randomness , penrose tiling , subdivision , square tiling , quasicrystal , topological entropy , tile , hexagonal tiling , topology (electrical circuits) , combinatorics , discrete mathematics , pure mathematics , computer science , geometry , statistics , archaeology , history , programming language , grid , art , visual arts

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

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