  We consider the range of possible dynamics of cellular automata (CA) on
two-sided beta-shifts $S_\beta$. We show that any reversible CA $F:S_\beta\to
S_\beta$ has an almost equicontinuous direction whenever $S_\beta$ is not
sofic. This has the corollary that non-sofic beta-shifts are topologically
direct prime. We also make some preliminary observations on direct topological
factorizations of beta-shifts that are subshifts of finite type.
