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?



You must log in to answer this question.

Browse other questions tagged .