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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.

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2 Answers 2

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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

Circles and rational functions

The case when the Julia set is CONTAINED in a smooth invariant curve was studied in this paper.

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  • $\begingroup$ 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). $\endgroup$
    – Luka Thaler
    Commented Nov 15, 2018 at 16:53
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The references given by Alexandre Eremenko came very useful when we were trying to find the answer to the quesiton (2). We proved that if $V\subset \widehat{\mathbb{C}}$ is a semi-analytic set of real dimension $1$ and if $\mu(V)>0$, then the julia set is contained in a circle (i.e. circle in $\widehat{\mathbb{C}}$). See Lemma 12 in the paper Reduced dynamical systems

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