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

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts S_β and its relation to direct topological factorizations. We show that any reversible CA F:S_β→S_β has an almost equicontinuous direction whenever S_β 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 of two nontrivial subshifts X and Y. We also give a simple criterion to determine whether S_nγ is conjugate to S_n×S_γ for a given integer n≥1 and a given real γ>1 when S_γ is a subshift of finite type. When S_γ is strictly sofic, we show that such a conjugacy is not possible at least when γ is a quadratic Pisot number of degree 2. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.