Mating Siegel and parabolic quadratic polynomials

TL;DR

This paper proves that certain quadratic polynomials with indifferent fixed points can be conformally mated with rational maps, establishing uniqueness and contributing to understanding Julia set connectivity.

Contribution

It demonstrates conformal mating between Siegel and parabolic quadratic polynomials for specific parameters, answering a question posed by Milnor.

Findings

Conformal mating is possible for bounded type Siegel and rational maps.

Mating is unique up to Möbius conjugacy.

Julia sets of certain Siegel rational maps are locally connected.

Abstract

Let $f_\theta(z)=e^{2\pi i\theta}z+z^2$ be the quadratic polynomial having an indifferent fixed point at the origin. For any bounded type irrational number $\theta\in\mathbb{R}\setminus\mathbb{Q}$ and any rational number $\nu\in\mathbb{Q}$, we prove that $f_\theta$ and $f_\nu$ are conformally mateable, and that the mating is unique up to conjugacy by a M\"{o}bius map. This gives an affirmative (partial) answer to a question raised by Milnor in 2004. A crucial ingredient in the proof relies on an expansive property when iterating certain rational maps near Siegel disk boundaries. Combining this with the expanding property in repelling petals of parabolic points, we also prove that the Julia sets of a class of Siegel rational maps with parabolic points are locally connected.