Julia set containing smooth curve

I have two realted questions.

Let $$R$$ be a rational function on $$\mathbb{C}$$ with degree at least 2. We denote by $$\mu$$ the measure of maximal entropy for $$R$$ and recall that the Julia set coincides with the support $$\mu$$.

(1) If the Julia set contains a smooth curve (real 1D analytic curve), what can we say about Julia set? (circle, line segment, cantor set of circles). I would expect it to be smooth.

(2) IF $$V$$ is a 1D real analyitic curve (or semianalyitic) then either $$\mu(V)=0$$ or else Julia set is contained in a real analyitic curve.

I was trying to find answers in the literature but unsuccessfully. I'll be happy for any comments or useful references.

• Dear Alexandre, thank you for your answer. I am aware of some of these results which have stronger assuptons but since this is not my main research area I optimistically thought that these things are known by now, and that I have just missed them. I am more concerned about the second question or the following version of it. Suppose $V$ is a smooth curve and that $J \cap V$ is not relatively open in $J=supp \mu$, then the measure $\mu(V) =0$. I would think that this is alway the case but for now I don't see how to prove it (maybe it is trivial). – user47862 Nov 15 '18 at 16:53