Let $f$ be a rational function on $\mathbf{C}$ with $\deg(f) \geq 2$, and denote by $J$ and $F$ its Julia and Fatou sets. Consider a connected component $A$ of $F$ which is not empty, forward-invariant ($f(A)=A$) and whose fronteer contains a repelling fixed point $x$. It can be shown that the fronteer $\partial A$ is also forward-invariant.
The question is the following. A classical result (see 1 cor 4.13) states that the backward orbit of $x$ ($\bigcup_{n\geq0} \, f^{-n}(x)$) is dense in $J$. But can we consider only the points in $\partial A$ ? Namely is \begin{equation} \bigcup_{n\geq0} (f^{-n}(x) \cap \partial A) \end{equation} dense in $\partial A$ ?
Remark : the question comes from a misunderstanding of proposition 23 proof in the following article : "On the asymptotic behaviour of analytic solutions of linear iterative functional equations" by E. Teufl.
1 J. Milnor, Dynamics in One Complex Variable, 3rd ed.