Direct topological factorization for topological flows

Tom Meyerovitch

Published 2014-07-31, updated 2014-11-27Version 2

We study the notion of \emph{direct factorization} for topological dynamical systems. This notion was considered the early 1980's by D. Lind for $\mathbb{Z}$-shifts of finite type. Here we consider more general situations, where the acting group $\mathbb{G}$ is countable but not necessarily equal to $\mathbb{Z}$. Also, we consider situations where the system is not a subshift of finite type. Direct factorizations for $\mathbb{G}$-shifts of finite type are considered, in particular when $\mathbb{G}=\mathbb{Z}^d$. We study direct factorizations for specific systems, and prove that the "$3$-colored-chessboard" and certain Dyck shifts are topologically direct-prime.