Thurston's skinning map played an important role in his original
hyperbolization theorem. His Bounded Image Theorem, in the acylindrical
case, gave control of the fixed-point problem on which the proof depends.
We generalize this theorem to a relative version which holds for manifolds
with cylinders, and along the way refine our understanding of how one
constructs compact cores with uniform topological and geometric control in
degenerating sequences of hyperbolic structures. This is joint work with J.
Brock, K. Bromberg and R. Canary.