rearrange $z \mapsto z^2 + c$

Question

The Mandelbrot Set:

$Z \mapsto Z² + C$ (or more precisely) $Z_{i+1} = Z_i ^2 + C$

Where $Z$ and $C$ are complex numbers.

Can this well-established equation be rearranged to determine things that would otherwise take a large or infinite number of iterations or tests with different values of $C$?

Such as the smallest bounding box that would contain all values for $C$, where $Z$ bails out in a given iteration.

Answer 1

$z$ and $c$ are complex numbers here, represented as points in the complex plane. The set is not the set of $z=z^2+c$, but the set of points $c$ such that the iteration $c, c^2+c, (c^2+c)^2+c, ((c^2+c)^2+c)^2+c \ldots$ does not go to infinity.

You can certainly use algebra to determine many of the features of the set. Each bulb has a root of one of these expressions in it, and the edge of the bulb is where the derivative has absolute value $1$.