Recall: the Mandelbrot set $M$ is defined to be the set of points $c\in \mathbb{C}$ such that the sequence of complex numbers $\{c, c^2+c, (c^2+c)^2 + c, \ldots \}$ is bounded in magnitude. (Recursively, the sequence is defined by $z_1 = c$ and $z_{k+1} = z_k^2 + c$.)

The Wikipedia page on the Mandelbrot set states that the intersection of $M$ with the real axis is precisely the closed interval $[-2,1/4]$.

What if we instead intersect the boundary $\partial M$ of the Mandelbrot set with the real axis? The figure here seems to suggest this intersection may be a discrete set of points, and that this is related to the logistic map. However I did not find the answer to this on either Wikipedia page, after browsing briefly. To summarize, I am wondering:

  • Can we say exactly what real points are in the boundary $\partial M$?

  • Is the intersection $\mathbb{R}\cap \partial M$ a discrete set of points?

  • If it is not discrete, is it dense in any open interval of $\mathbb{R}$?

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    $\begingroup$ Where there's an attractive fixpoint, there's a gap (i.e., an interior point of $M$); where there's bifurcarion, there's an isolated point. But inberween there are regions of chaos, literally; I'd guess the boundary is dense there. We can largely view $M$ as obtained from a disk by pinching together rational boundary points - so we might find some number-theoretic properties that guide us to the answer to this question $\endgroup$ – Hagen von Eitzen Jun 8 '17 at 5:14
  • $\begingroup$ I think there should be an infinite number of "baby Mandelbrots" between any two chaotic boundary points. $\endgroup$ – Sheldon L Jun 8 '17 at 19:03

The only points in the interior of the Mandelbrot set are those inside the main cardioid, or inside a bulb off the main cardioid, or inside a baby mandelbrot. This assumes the hyperbolicity conjecture that there are no queer/ghost interior components; which is unproven, but it is proven on the real axis. All the other points are on the boundary of the Mset. You can't have a "dense" set of points that are both on the boundary and on the real axis; all such points are discreet points, with a hyperbolic region (baby Mandelbrot) arbitrarily close by. This question is related and gives a lot more detail. M-set interior point probability on the real axis

The hyperbolicity conjecture is implied by the MLC (Mandelbrot locally connected conjecture). https://mathoverflow.net/questions/95701/the-deep-significance-of-the-question-of-the-mandelbrot-sets-local-connectednes The "Density of hyperbolicity is known if one restricts to real c". I think it was first proven at the real axis by Lyubich.

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    $\begingroup$ could you provide more detail or references for some of these terms? I'm not familiar with the hyperbolicity conjecture or queer/ghost components. $\endgroup$ – Harry Richman Jul 28 '17 at 4:06
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    $\begingroup$ mathoverflow.net/questions/95701/… The MLC conjecture implies the hyperbolicity conjecture, and "Density of hyperbolicity is known if one restricts to real c". I think it was first proven at the real axis by Lyubich. $\endgroup$ – Sheldon L Jul 30 '17 at 15:35

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