Hyperbolic-parabolic deformations of rational maps

Abstract

We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in 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.

The landing of parameter rays at non-recurrent critical portraits

“…where E 1 and E 2 are disjoint full continua in V and A(E 1 , E 2 ) := C \ (E 1 ∪ E 2 ). The following result is an equivalent variation of [5,Theorem 8.1].…”

“…Following [5], the modulus difference distortion of an annulus is used to estimate the distortion between a univalent map and a Möbius transformation. Let V ⊆ C be an open set and φ : V → C be a univalent map.…”

“…Our definition of m-nested, λ-scattered disk systems is a minor generalization of the concept of s(r)-nested, λ-scattered disk systems given in [5,Section 8.2].…”

“…Comparing with the original concept of s(r)-nested disk systems given in [5], we remark that if {D x } x∈X is an s(r)-nested disk system, then D y must avoid x if D y ⊆ D x . However, our definition of m-nested disk systems does not have this restriction.…”

“…In [3], the authors give an alternative proof of the Douady-Hubbard ray-landing theorem for quadratic Misiurewicz polynomials by using a new type of distortion theorem given in [5]. Based on this distortion theorem and with a similar idea mentioned in [3] and [5,Section 8], we can generalize the ray-landing theorem to non-recurrent polynomials with any degree d ≥ 2.…”