Complementary components to the cubic principal hyperbolic domain

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 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 queer type (then a specially chosen critical point of $f\in \mathcal {W}$ belongs to $J(f)$, the set $J(f)$ has positive Lebesgue measure, and it carries an invariant line field).