In this paper, we develop a theory on the degenerations of Blaschke products B d to study the boundaries of hyperbolic components. We give a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component H d containing z d . We show the closure H d is not a topological manifold with boundary for d ≥ 4 by constructing self-bumps on its boundary.
In this article, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.
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
Q
B
d
\mathcal {QB}_d
, making progress towards the analogues of Thurston’s compactness theorem for acylindrical
3
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.
We will show the Mandelbrot set M is locally conformally inhomogeneous: the only conformal map f defined in an open set U intersecting ∂M and satisfying f (U ∩ ∂M ) ⊂ ∂M is the identity map.The proof uses the study of local conformal symmetries of the Julia sets of polynomials: we will show in many cases, the dynamics can be recovered from the local conformal structure of the Julia sets.
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