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.