Let $f:\mathbb C\to \mathbb C$ be a rational map and let $J(f)$ and $F(f)$ denote the Julia and Fatou sets of $f$, respectively.
Let $\mathcal S$ be the set of all boundaries of Fatou components. Assume that each member of $\mathcal S$ is a simple closed curve, and that $J(f)$ is locally connected. Some examples of rational Julia sets with this property are the circle (e.g. $f(z)=z^2$), Sierpinski carpet ($f(z)=z^2+\frac{1}{z^2}$), and Sierpinski triangle ($f(z)=z^2-\frac{1}{z}$).
Question. For all (distinct) $S_1,S_2\in \mathcal S$ is it necessarily true that $S_1\cap S_2$ is finite? Is the cardinality of all such intersections bounded by a finite constant $n$?
Of course this is trivial if the Julia set is a circle. For the Sierpinski carpet the intersections are empty. And for the Sierpinski triangle each intersection has at most 3 points.
