Consider the classical Julia set $J_c$ associated with $z\mapsto z^2+c$. Then the image of $J_c$ under the inverse map $z\mapsto \pm \sqrt{z-c}$ lie in $J_c$, since $J_c$ is completely invariant. Now, let $H_c$ be the convex hull of $J_c$.

Is it true that the image of $H_c$ under the map $z\mapsto \pm \sqrt{z-c}$ lie in $H_c$?

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.

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 $f^{-1}(H_f)\subset H_f$.