The dynamics group of an asynchronous cellular automaton (ACA) relates
properties of its long term dynamics to the structure of Coxeter groups. The
key mathematical feature connecting these diverse fields is involutions.
Group-theoretic results in the latter domain may lead to insight about the
dynamics in the former, and vice-versa. In this article, we highlight some
central themes and common structures, and discuss novel approaches to some open
and open-ended problems. We introduce the state automaton of an ACA, and show
how the root automaton of a Coxeter group is essentially part of the state
automaton of a related ACA.Comment: 10 pages, 4 figure
