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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.
| Author details: | Johannes Guetschow, Vincent Nesme, Reinhard F. Werner |
|---|---|
| ISSN: | 1557-5969 |
| Title of parent work (English): | Journal of cellular automata |
| Publisher: | Old City Publishing Science |
| Place of publishing: | Philadelphia |
| Publication type: | Article |
| Language: | English |
| Year of first publication: | 2012 |
| Publication year: | 2012 |
| Release date: | 2017/03/26 |
| Tag: | abelian group; fractal; linear cellular automaton; self-similarity; substitution system |
| Volume: | 7 |
| Issue: | 2 |
| Number of pages: | 31 |
| First page: | 83 |
| Last Page: | 113 |
| Funding institution: | Deutsche Forschungsgemeinschaft [Forschergruppe 635]; EU; Erwin Schrodinger Institute; Rosa Luxemburg Foundation |
| Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie |
| Peer review: | Referiert |
