标题： 阿贝尔群上的细胞自动机的分形结构 显示英文标题

标题： The fractal structure of cellular automata on Abelian groups

摘要： 众所周知，某些细胞自动机的时空图具有分形结构：例如，模2的帕斯卡三角形生成一个谢尔宾斯基三角形。 解释细胞自动机时空图的分形结构是一个被广泛研究的主题，但几乎所有的结果都围绕着一类特殊的自动机，其典型特征包括不可逆性、具有环结构的字母表、全局演化为环同态，以及一种称为（弱）p-费马的性质。 本文研究的自动机类别不具有这些特性。 它们的单元结构较弱，因为它不带有乘法，并且它们远非p-费马，甚至弱形式也不是。 然而，它们确实会产生分形时空图，我们将解释原因和方式。 显示英文摘要

摘要： It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much 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 fractal spacetime diagrams, and we explain why and how.