Mandelbrot Set - Predict which value of c will give bounded results?

Question

I have been looking into the Mandelbrot set a little bit lately, and I have a question.

Given the equation: $$z(n+1) = (zn)^2 + c$$ where $c$ is a complex number of the form $a+bi$ is there an easy way of seeing if that complex number is bounded and apart of the Mandelbrot Set?

Thanks

Answer 1

In general, no. That is what make the Mandelbrot set interesting. You can define regions that are clearly in, like the main cardioid, the circles attached to it, the mini-brots elsewhere, etc. You can define regions which are clearly out, like $|z| \gt 2$. In between is a mess........