Complementary components to the cubic Principal Hyperbolic Domain

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 $f{\mathrm{\mid }}_V$ 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).