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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!

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    $\begingroup$ A fixed point that attracts supper? That would be interesting. $\endgroup$
    – Gerry Myerson
    Commented Mar 13, 2018 at 11:23

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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.

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