Mandelbrot Set- definitive point testing

1. Nov 19, 2008

Savant13

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. Nov 19, 2008

Savant13

Should I have posted this somewhere else?

3. Nov 20, 2008

gel

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
4. Nov 24, 2008

Savant13

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.

5. Nov 24, 2008

Hurkyl

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