Consider the classical Julia set $J_f$ associated with $f(z)=z^2+c$.
Since $J_c$ is completely invariant,
we know that $f^{-1}(J_f) \subseteq J_f$.
Now, let $H_f$ be the convex hull of $J_f$.

**Is it true that  $f^{-1}(H_f) \subseteq H_f$?**

I have done some basic computer experiments, and it seem to hold for $c \in [0,1]^2 \subset \mathbb{C}$. Moreover, I suspect that the natural generalization of the statement above might hold for all polynomial maps. However, I have examples with rational maps where the statement is not true.


*As an example,* consider $f(z)=z^3-iz + 0.2 + 0.4i$.
The blue points is the Julia set $J_f$ associated with $f$. The shaded region is the convex hull $H_f$ of the Julia set.
Taking a uniform square grid $G$ on $H_f$, and  plotting 
the points $f^{-1}(G)$ gives the black dots.
As we can see, it is reasonable to guess that $f^{-1}(H_f)\subset H_f$. 
[![Julia set of x^3-ix+0.2+0.4i][1]][1]


  [1]: https://i.sstatic.net/mR9tP.png