An organizing principle in the study of the Jacobian Conjecture

Let $\Omega$ be an irreducible component of the locus of polynomial maps $ F:\mathbb C^n \to \mathbb C^n$ satisfying $\text{\rm deg} F\leq k$ and $\text{det}DF=1$. It is shown that either $\Omega \subset \text{\rm Aut} (\mathbb C^n)$, as claimed by the Jacobian Conjecture, or the {\em general} $F\in \Omega$ is not an automorphism.