Iterations of rational functions: which hyperbolic components contain polynomials?

Let ${\displaystyle H^d}$ be the set of all rational maps of degree d ≥ 2 on the Riemann sphere, expanding on their Julia set. We prove that if ${\displaystyle f\in H^d}$ and all, or all but one, critical points (or values) are in the basin of immediate attraction to an attracting fixed point then there exists a polynomial in the component H(f) of ${\displaystyle H^d}$ containing f. If all critical points are in the basin of immediate attraction to an attracting fixed point or a parabolic fixed point then f restricted to the Julia set is conjugate to the shift on the one-sided shift space of d symbols. We give exotic} examples of maps of an arbitrary degree d with a non-simply connected completely invariant basin of attraction and arbitrary number k ≥ 2 of critical points in the basin. For such a map ${\displaystyle f\in H^d}$ with k