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 S β  Sβ. We show that any reversible CA F:S β →S β F:Sβ→Sβ has an almost equicontinuous direction whenever S β Sβ 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 X×Y X×Y 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.