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Self-similarity of cellular automata on abelian groups

  • It is well known that the spacetime diagrams of some cellular automata have a self-similar fractal structure: for instance Wolfram's rule 90 generates a Sierpinski triangle. Explaining the self-similarity of the spacetime diagrams of cellular automata is a well-explored topic, but virtually all of the results revolve around a special class of automata, whose typical features include irreversibility, an alphabet with a ring structure, a global evolution that is a ring homomorphism, and a property known as (weakly) p-Fermat. The class of automata that we study in this article has none of these properties. Their cell structure is weaker, as it does not come with a multiplication, and they are far from being p-Fermat, even weakly. However, they do produce self-similar spacetime diagrams, and we explain why and how.

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Author:Johannes Guetschow, Vincent Nesme, Reinhard F. Werner
ISSN:1557-5969 (print)
Parent Title (English):Journal of cellular automata
Publisher:Old City Publishing Science
Place of publication:Philadelphia
Document Type:Article
Year of first Publication:2012
Year of Completion:2012
Release Date:2017/03/26
Tag:abelian group; fractal; linear cellular automaton; self-similarity; substitution system
First Page:83
Last Page:113
Funder:Deutsche Forschungsgemeinschaft [Forschergruppe 635]; EU; Erwin Schrodinger Institute; Rosa Luxemburg Foundation
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
Peer Review:Referiert