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Using the Baire category theorem, we may show that most simple closed curves satisfy the following property: any segment between an interior point and an exterior point of the curve intersects the curve in infinitely many points. First, I am looking for a printed reference of this result.

Baire category arguments are not very explicit. I am also looking for a concrete example of such a curve.

More precisely, in the quadratic family $z \mapsto z^2 +c$, is there an explicit parameter such that the associated Julia set has a Jordan curve as its boundary and such that this curve satisfies the aforementioned property?

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  • $\begingroup$ Ok, I've talked to an expert, and they suggest that if $c$ lies inside the main cardioid of the Mandelbrot set, and if no multiplier of of any periodic point of $f_c$ is real, then $J_c$ should have the desired property. Thus for $c$ off of a countable union of real analytic curves we win... $\endgroup$
    – Sam Nead
    Commented Oct 25, 2019 at 17:14

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