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Conformal maps from the complement of the Mandelbrot set to the disk are well known. Also, it is known that there exists a surjective conformal map from the interior of the Mandelbrot set to the disk by the Riemann mapping theorem.

Is an explicit example of a conformal map from the Mandelbrot set to the unit disk known?

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closed as off-topic by Xander Henderson, A. Goodier, Rhys Steele, zz20s, user223391 Apr 4 '18 at 2:46

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    $\begingroup$ Does the Riemann mapping theorem apply? Is the Mandelbrot set open? Is the interior of the Mandelbrot set simply connected? $\endgroup$ – Claude Apr 2 '18 at 19:47
  • $\begingroup$ @Claude Oh, good point. I forgot about the openness requirement. $\endgroup$ – PyRulez Apr 2 '18 at 22:27
  • $\begingroup$ @Claude You should be able to do it piecewise on the connected components of the interior, though. $\endgroup$ – PyRulez Apr 3 '18 at 0:16
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For the revised question suggested in the comment:

you should be able to do it piecewise on the connected components of the interior

You can use the derivative of the limit cycle of points in hyperbolic components as a coordinate in the unit disc. I believe the mapping is conformal, but I don't have a proof handy...

interior coordinates

Practically, you would check candidate periods $p$ where $|z_p|$ reaches a new minimum in iterations of $z_{n + 1} = f_c(z_n)$. Use Newton's method to solve $z = f_c^p(z)$, using $z_p$ as initial guess. If $\left|\frac{\partial}{\partial z} f_c^p(z)\right| \le 1$ (evaluated at the root you found), then $c$ is inside a hyperbolic component of period $p$, and $w = \frac{\partial}{\partial z} f_c^p(z)$ is mapped to the unit disc. I have no proof of correctness of this algorithm, but it seems to work...

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