# Which real points are in the boundary of the Mandelbrot set?

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 $$\mathbb{R}$$?

• 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 – Hagen von Eitzen Jun 8 '17 at 5:14
• I think there should be an infinite number of "baby Mandelbrots" between any two chaotic boundary points. – Sheldon L Jun 8 '17 at 19:03