We study the closure of the cubic Principal Hyperbolic Domain and its intersection Pλ with the slice Fλ of the space of all cubic polynomials with fixed point 0 defined by the multiplier λ at 0. We show that any bounded domain W of Fλ \Pλ consists of J-stable polynomials f with connected Julia sets J(f) and is either of Siegel capture type (then f ∈ W has an invariant Siegel domain U around 0 and another Fatou domain V such that f|V is two-to-one and fk (V ) = U for some k > 0) or of queer type (then a specially chosen critical point of f ∈ W belongs to J(f), the set J(f) has positive Lebesgue measure, and it carries an invariant line field).
