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!



Does the boundary of immediate basin contain a fixed point?
Yes. This was proved by Fatou,
Sur les equations fonctionnelles, Bull SMF, 48 (1920) on p. 81.


See also:


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. 


