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 polynomial can be written as follows:

$$P_c^n(z)=\Phi(n+\Psi(z))$$

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