Hyperbolic-parabolic deformations of rational maps

We develop a Thurston-like theory to characterize geometrically finite rational maps, and then apply it to study pinching and plumbing deformations of rational maps. We show that under certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.