Mandelbrot Set- definitive point testing

[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.

 

[Savant13]

Should I have posted this somewhere else?

 

[gel]

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

In the Blum-Shub-Smale model of real computation, the Mandelbrot set is not computable, but its complement is computably enumerable. However, many simple objects (e.g., the graph of exponentiation) are also not computable in the BSS model. At present it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis, which correspond more closely to the intuitive notion of "plotting the set by a computer." Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true.

 

[Savant13]

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

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.

 

[Hurkyl]

c can get very large as long as it converges finitely eventually.

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