Direct topological factorization for topological flows

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of non-trivial actions. In the early 1980s, D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. In particular, we study direct factorizations of subshifts of finite type over $\mathbb{Z}^{d}$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive $\mathbb{G}$-action must be finite, but an example is provided of a non-expansive $\mathbb{Z}$-action for which there is no finite direct-prime factorization. The question about existence of direct-prime factorization of expansive actions remains open, even for $\mathbb{G}=\mathbb{Z}$.