Mandelbrot Set - Generalizations

Generalizations

Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions

For integer d, these sets are connectedness loci for the Julia sets built from the same formula. The full cubic connectedness map has also been studied; here one considers the two-parameter recursion, whose two critical points are the complex square roots of the parameter k. A point is in the map if either critical point is stable.

For general families of holomorphic functions, the boundary of the Mandelbrot set generalizes to the bifurcation locus, which is a natural object to study even when the connectedness locus is not useful.