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.



Julia set containing smooth curve
I am not surprised that you found nothing in the literature.


On question 1: it has been studied under a much stronger assumption that the Julia set contains a smooth INVARIANT curve. Under some additional conditions, it was proved that such a curve must be algebraic. And there are such algebraic curves, other than circles.
But without additional conditions this is not true. Nothing special can be said about Julia sets in these cases.


[Invariant curves and semiconjugacies of rational functions](https://arxiv.org/abs/1110.6552)


[Circles and rational functions](https://mathoverflow.net/questions/103949/circles-and-rational-functions)


The case when the Julia set is CONTAINED in a smooth invariant curve was studied in [this paper](http://www.math.purdue.edu/~eremenko/dvi/real-2.pdf).


