On geometrically finite degenerations II: convergence and divergence

In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits such as geometrically maps on a tree of Riemann spheres. prove the boundedness hyperbolic with Sierpinski carpet Julia set and give criteria convergence quasi-Blaschke products Q mathvariant="script">B d \mathcal {QB}_d , making progress towards analogues Thurston’s compactness theorem acylindrical alttext="3"> 3 encoding="application/x-tex">3 -manifold double limit quasi-Fuchsian groups in complex dynamics. appendix, apply results to show existence certain polynomial matings.