# Simple ways for checking if a point is not in a multibrot set

As is well-known, the Mandelbrot set (M-set for short) can be defined by considering the family of functions $$f_c(z)=z^2+c$$ for $$c\in\mathbb{C}$$, iterating them for each $$c$$ to obtain sequences $$z_{n+1}=f_c(z_n)$$ starting with $$z_0=0$$, and checking whether or not $$|z_n|$$ tend to infinity -- if they don't, $$c$$ is in the M-set, if they do, $$c$$ isn't in the M-set.

Not only that but there is a well-known condition for checking the latter: if $$|z_n|>2$$ for some $$n$$, then $$c$$ is not in the M-set.

Question: Is there an analogous condition for checking when points are not in a multibrot set, i.e. when one considers $$f_c(z)=z^d+c$$ for $$d>2$$ and repeats the above considerations? Is there some number $$R(d)$$ dependent on $$d$$ such that $$|z_n|>R(d)$$ for some $$n$$ implies that the corresponding choice of $$c$$ is not in the given multibrot set?

• Yes - $R(d)=2$ works for all $d$ and the proof is exactly the same as in the $d=2$ case. Commented Oct 28, 2019 at 12:53
• Well that's convenient. Thanks! Commented Oct 28, 2019 at 13:11
• fractalforums.org/fractal-mathematics-and-new-theories/28/… Commented Oct 29, 2019 at 21:23