Misiurewicz maps unfold generically (even if they are critically non-finite)

We show that in normalized families of polynomial or rational maps, Misiurewicz maps (critically finite or infinite) unfold generically. For example, if ${\displaystyle f_{{\lambda }_0}}$ is critically finite with non-degenerate critical point ${\displaystyle c_1\left({\lambda }_0\right),...,c_n\left({\lambda }_0\right)}$ such that ${\displaystyle f_{{\lambda }_0}^{k_i}\left(c_i\left({\lambda }_0\right)\right)=p_i\left({\lambda }_0\right)}$ are hyperbolic periodic points for i = 1,...,n, then  IV-1. Age impartible......................................................................................................................................................................... 31  ${\displaystyle \lambda \mapsto (f_{\lambda }^{k_1}\left(c_1(\lambda )\right)-p_1(\lambda ),...,f_{\lambda }^{k_{d-2}}\left(c_{d-2}(\lambda )\right)-p_{d-2}(\lambda ))}$ is a local diffeomorphism for λ near ${\displaystyle {\lambda }_0}$. For quadratic families this result was proved previously in {DH} using entirely different methods.