"Necks" of the Mandelbrot set

Question

I'm looking at some "neck" points of the Mandelbrot set where there is just one point connecting a part of the set to another, such as the point (-0.75,0). I'm currently interested in finding out the precise values of the 2 necks connecting bulbs with the same real part on the top and bottom to the main body of the set. Zooming in on the set using WolframAlpha tells me that that point is around (-0.125,0.65), but is that the exact value of such a point or do the coordinates of it involve irrational numbers?

Answer 1

In general these points are solutions $c$ to simultaneous equations $$f_c^n(z) = z$$ $$(f_c^n)'(z) = \exp\left(2\pi i \frac{p}{q}\right)$$ where $$f_c^0(z)=z$$ $$f_c^{n+1}=f_c^n(z)^2+c$$ They are parabolic points where an attractive period $n$ orbit in the "parent" bifurcates into a repelling period $n$ orbit and an attractive period $qn$ orbit in the "child"

For example, $n=2$ gives $$c = \frac {\exp\left(2\pi i \frac{p}{q}\right)}{4}-1$$ but in general the solutions are irrational algebraic numbers without nice closed forms.

Newton's method in two complex variables can find numerical solutions given a nearby guess $c_0, z_0$ ($z_0=0$ is usually ok).