Complementary components to the cubic Principal Hyperbolic Domain

TL;DR

This paper investigates the structure of the cubic Principal Hyperbolic Domain, identifying the types of stable polynomial domains outside it, including Siegel capture and queer types, with implications for Julia set properties.

Contribution

It characterizes the components outside the cubic Principal Hyperbolic Domain, distinguishing between Siegel capture and queer types with detailed dynamical properties.

Findings

Bounded domains outside the principal hyperbolic domain are composed of J-stable polynomials.

Such domains are either Siegel capture type or queer type.

Queer type domains have Julia sets with positive Lebesgue measure and invariant line fields.

Abstract

We study the closure of the cubic Principal Hyperbolic Domain and its intersection $\mathcal{P}_\lambda$ with the slice $\mathcal{F}_\lambda$ of the space of all cubic polynomials with fixed point $0$ defined by the multiplier $\lambda$ at $0$. We show that any bounded domain $\mathcal{W}$ of $\mathcal{F}_\lambda\setminus\mathcal{P}_\lambda$ consists of $J$-stable polynomials $f$ with connected Julia sets $J(f)$ and is either of \emph{Siegel capture} type (then $f\in \mathcal{W}$ has an invariant Siegel domain $U$ around $0$ and another Fatou domain $V$ such that $f|_V$ is two-to-one and $f^k(V)=U$ for some $k>0$) or of \emph{queer} type (then at least one critical point of $f\in \mathcal{W}$ belongs to $J(f)$, the set $J(f)$ has positive Lebesgue measure, and carries an invariant line field).