On geometrically finite degenerations II: convergence and divergence

Abstract

In this paper, we study quasi post-critically finite degenerations for rational maps. We construct a limit for such degeneration as a geometrically finite rational map on a finite tree of Riemann spheres. We prove the boundedness for such degenerations of hyperbolic rational maps with Sierpinski carpet Julia set and characterize the convergence for quasi-Blaschke products QB d . These are natural analogs of Thurston's compactness theorem for acylindrical 3-manifold and the double limit theorem for quasi-Fuchsian groups. In the appendix, we apply such convergence results to show the existence of certain polynomial matings. Contents1. Introduction 1 2. Blaschke model spaces B S 3. Degenerations in B S 4. Limits of quasi post-critically finite degenerations 5. Convergence of quasi-invariant forests 6. Boundedness of Sierpinski carpet rational maps 7. A double limit theorem for QB d Appendix A. Applications on polynomial matings References