@article{GuetschowNesmeWerner2012, author = {Guetschow, Johannes and Nesme, Vincent and Werner, Reinhard F.}, title = {Self-similarity of cellular automata on abelian groups}, series = {Journal of cellular automata}, volume = {7}, journal = {Journal of cellular automata}, number = {2}, publisher = {Old City Publishing Science}, address = {Philadelphia}, issn = {1557-5969}, pages = {83 -- 113}, year = {2012}, abstract = {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.}, language = {en} }