Hyperbolic-parabolic deformations of rational maps

TL;DR

This paper develops a Thurston-like framework to analyze deformations of rational maps, demonstrating convergence properties of pinching paths and characterizing geometrically finite maps with parabolic points.

Contribution

It introduces a new theory for geometrically finite rational maps and explores the dynamics of pinching and plumbing deformations with convergence results.

Findings

Pinching paths converge uniformly under certain conditions.

Quasiconformal conjugacies approach semi-conjugacies.

Geometrically finite rational maps with parabolic points are limits of pinching deformations.

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.