We know that there exists a polynomial the Fatou set $F(P)$ is connected, which is just an attracting basin for infinity.

I have a question: Given a rational function $R$ such that $F(R)$ is connected, is this always true that $F(R)$ is just a attracting basin?

If the question is not, I wondered whether there exists a rational function $R$, such that $F(R)$ is connected and $F(R)$ is a completely invariant parabolic petal.

Further question: Is it possible to classify all rational maps $R$ with exactly one one Fatou component.

Any comments and remarks will be appreciated.