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Mandelbrot Set- definitive point testing

  1. Nov 19, 2008 #1
    For those of you not familiar with the mandelbrot set, it is the set of all complex numbers c for which the following transform remains finite for an infinite number of iterations:

    z --> z^2 + c

    z is 0 for the first iteration

    My question is this: How can I conclusively determine whether a number is part of the Mandelbrot Set? I am not looking for an approximation, I can do that on my own.
    Last edited: Nov 19, 2008
  2. jcsd
  3. Nov 19, 2008 #2
    Should I have posted this somewhere else?
  4. Nov 20, 2008 #3


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    maybe no-one knows the answer?

    I've never studied the Mandlebrot set, but can say the following

    - if c is not in the Mandlebrot set then the sequence z_n will eventually diverge, and you can prove this with certainty just by evaluating the sequence accurately enough until z becomes large.

    - for certain values of c the sequence z_n will be eventually be periodic and c will be in the Mandlebrot set, and you could compute these values as solutions of polynomials.

    - for more general points in the Mandlebrot set I doubt that there is such an algorithm.

    edit: Wikipedia seems to back up what I just said

    Last edited: Nov 20, 2008
  5. Nov 24, 2008 #4
    Evalutating until c becomes large doesn't work except as an approximation. c can get very large as long as it converges finitely eventually. You also cannot prove that c does not converge to a finite number or range through such evaluation.
  6. Nov 24, 2008 #5


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    Staff Emeritus
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    If c is sufficiently large, iterating must diverge to infinity.
    If z becomes sufficiently large, iterating must diverge to infinity.
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