  In this paper we consider families of holomorphic maps defined on subsets of
the complex plane, and show that the technique developed in \cite{LSvS1} to
treat unfolding of critical relations can also be used to deal with cases where
the critical orbit converges to a hyperbolic attracting or a parabolic periodic
orbit. As before this result applies to rather general families of maps, such
as polynomial-like mappings, provided some lifting property holds. Our Main
Theorem states that either the multiplier of a hyperbolic attracting periodic
orbit depends univalently on the parameter and bifurcations at parabolic
periodic points are generic, or one has persistency of periodic orbits with a
fixed multiplier.
