Mandelbrot polynomial

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Mandelbrot polynomial is special kind of quadratic polynomial, written in form $P_c(z)=z^2+c$ whrere $c$ is parameter. Usially, it is assumed to be a complex number.

Mandelbrot set

The Mandelbrot polynomial is used to define the Mandelbrot set

$M = \left\{c\in \mathbb C : \exists s\in \mathbb R, \forall n\in \mathbb N, |P_c^n(0)| \le s \right\}$

SuperMandelbrot

Superfunction $\Phi$ for the transfer function $P_c$ can be constructed from the Superfunction $F$ of the Logistic operator:

$\Phi(z)=p(F(z))$

where

$p(z) = r z -r/2$,

$r\displaystyle =\frac{1}{2}+\sqrt{\frac{1}{4}+c}$

Similarly, the corresponding Abel function $\Psi=\Phi^{-1}$ can be expressed through the Abel function $G$ of the Logistic operator,

$\Psi(z) = G(q(z))$ where $\displaystyle q(z)=\left(\frac{1}{2}-z\right)/r$.

Then, following the general rule, through the superfunction $\Phi$ and the Abel Function $\Psi$, the $n$th iteration of the Mandelbrot operator can be written as follows: $P_c^n(z)=\Phi(n+\Psi(z))$

In this expression, number of iterations $n$ has no need to be integer; the Mandelbrot operator can be iterated arbitrary (even complex) number of times.