Dynamics of Cellular Automata on Beta-Shifts and Direct Topological Factorizations

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts [Formula: see text]. We show that any reversible CA [Formula: see text] has an almost equicontinuous direction whenever [Formula: see text] is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations [Formula: see text] of two nontrivial subshifts X and Y. We also make some preliminary observations on direct topological factorizations of beta-shifts that are subshifts of finite type.