Direct topological factorization for topological flows

TL;DR

This paper investigates conditions under which topological group actions, especially subshifts of finite type, can be decomposed into direct products, highlighting differences between expansive and non-expansive actions and presenting new examples.

Contribution

It provides new insights into direct factorizations of multidimensional subshifts, especially for non-expansive actions, and explores the open problem of prime factorization existence.

Findings

Expansive actions have finite direct factorizations.

Non-expansive actions may lack finite prime factorizations.

Open problem remains for expansive actions, even over algf3s.

Abstract

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of a nontrivial actions. In the early 1980's D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. We study in particular 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 a 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 $\mathbb{G}=\mathbb{Z}$.