to relate the monodromy stretch factors and rank of kernels for all symmetric rays in the same component. In the third paper [@DKL17c], they conclude this analysis by showing that being irreducible and atoroidal is an invariant of a component of the BNS-invariant; that is, if $\phi_1: K_1 \to K_1$ and $\phi_2: K_2 \to K_2$ are free group automorphisms corresponding to symmetric rays in the same component of $\mathcal C(G)$, then $\phi_1$ is irreducible and atoroidal if and only if $ \phi_2 $ is too. Since this result relied heavily on the analysis of the semi-flow for a component of the BNS-invariant, the technique cannot be extended to work for symmetric rational rays in separate components. Furthermore, their result does not apply to any asymmetric rational rays, i.e., it does not apply to nonsurjective injective endomorphisms. Theorem \[nongeomcomm\] addresses both of these concerns.

Masai-Mineyama have also proven a different special case of Theorem \[nongeomcomm\] that they call [*fibered commensurability*]{} [@MM17]: suppose $\phi: F \to F$ and $\psi: F' \to F'$ are free group automorphisms and let $K \le F, K' \le F'$ be finite index subgroups with an isomorphism $\epsilon: K \to K'$ such that there are the outer classes 