Rigidity of Simple Parabolic Maps

Abstract

We prove that two simple parabolic rational maps with conformally conjugate completely invariant parabolic Fatou domains are globally conformally conjugate. The proof combines quasiconformal rigidity with the absence of invariant line fields for simple parabolic maps. Together with our previous existence theorem, the rigidity result gives existence and uniqueness of simple parabolic models for infinitely-connected fixed parabolic Fatou domains. We further obtain a rational-map analogue of the Goldberg--Milnor perturbation conjecture.

33 pages, 4 figures