  We say that a finite asynchronous cellular automaton (or more generally, any
sequential dynamical system) is pi-independent if its set of periodic points
are independent of the order that the local functions are applied. In this
case, the local functions permute the periodic points, and these permutations
generate the dynamics group. We have previously shown that exactly 104 of the
possible 256 cellular automaton rules are pi-independent. In this article, we
classify the periodic states of these systems and describe their dynamics
groups, which are quotients of Coxeter groups. The dynamics groups provide
information about permissible dynamics as a function of update sequence and, as
such, connect discrete dynamical systems, group theory, and algebraic
combinatorics in a new and interesting way. We conclude with a discussion of
numerous open problems and directions for future research.
