On geometrically finite degenerations II: convergence and divergence

Luo, Yusheng

doi:10.1090/tran/8597

On geometrically finite degenerations II: convergence and divergence
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Trans. Amer. Math. Soc. 375 (2022), 3469-3527 Request permission

Abstract:

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

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