Let $f$ be a rational map of degree $d\geq 2$, and $B$ is a simply connected immediate basin of an supper-attracting fixed point of $f$. I want to know whether there exists a fixed point of $f$ contained in $\partial B$. Any hint will be welcome! Or some similar result can be given!
Yes. This was proved by Fatou, Sur les equations fonctionnelles, Bull SMF, 48 (1920) on p. 81.
MR1295160 F. Przytycki, A. Zdunik, Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps: geometric coding trees technique, Fund. Math. 145 (1994), no. 1, 65–77.