An atlas of limit set dynamics for asynchronous elementary cellular automata

In this paper we provide an overview of the possible ω-limit set structures for 104 of the 256 asynchronous elementary cellular automata over the circle graph on n vertices. We consider only fixed, sequential updates where the update sequence is given by a permutation of the vertices, that is, the class of sequential dynamical systems. The ECA rules covered are precisely the π-invariant rules, that is, the rules for which the set of periodic points does not depend on the permutation update sequence. This paper reviews existing work on π-invariance and cycle-equivalence, and provides and atlas of the possible limit set structures up to topological conjugation.