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Findding points within epsilon := 1/10^100 of the boundary of the Mandelbrot set: easy, or hard?
Posted:
Oct 7, 2013 7:31 AM


After looking at a video zooming in by 10^275 on the Mandelbrot set, it's a weird set.
cf.: < https://www.youtube.com/watch?v=0jGaio87u3A > .
So, suppose A is a point "close enough" (say within 0.05 roughly) of the centroid of the Mandelbrot set.
Let theta be a "random" angle in [0, 2pi[.
The L_{theta} be the ray from A at angle theta to the Real axis, going counterclockwise, from radius= 0 to radius = 100. It is a closed set; L_{theta} is a compact set in R^2, or C.
If M is the Mandelbrot set, M is compact. So is L_{theta} /\ M is also compact, and the distance function d:C > R given by d(X) =  X  A  is continuous on C.
Therefore, d() attains its maximum at a point P_{theta}. The point P_{theta} is both in M and in L_{theta}. Any open neighborhood of P_{theta} will contain a point in L_{theta} that is not in M. Therefore, P_{theta} is on the boundary of M.
epsilon := 1/10^100
For a pseudorandom theta, such as log(5) radians, how easy or hard is it to determine a point Y in C such [alternatively: M] such that  Y  P_{theta}  < epsilon ?
If epsilon = 1/10^100 is too easy, then we can ask same for epsilon = 1/10^10000 or even epsilon = 1/10^1000000 .
David Bernier
 Let us all be paranoid. More so than no such agence, Bolon Yokte K'uh willing.



