Flow equivalence of sofic beta-shifts

The Fischer, Krieger, and fiber product covers of sofic beta-shifts are constructed and used to show that every strictly sofic beta-shift is $2$-sofic. Flow invariants based on the covers are computed, and shown to only depend on an single integer easily determined from the $\beta$-expansion of 1. It is shown that any beta-shift is flow equivalent to a beta-shift given by some $1< \beta < 2$, and concrete constructions lead to further reductions of the flow classification problem. For each sofic beta-shift, there is an action of $\mathbb Z/2\mathbb Z$ on the edge shift given by the fiber product, and it is shown precisely when there exists a flow equivalence respecting these $\mathbb Z/2\mathbb Z$-actions. This opens a connection to ongoing efforts to classify general irreducible 2-sofic shifts via flow equivalences of reducible SFTs equipped with $\mathbb Z/2\mathbb Z$-actions.