Mandelbrot polynomial

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.

Keywords
Iteration , Logistic operator , Mandelbrot set , Superfunction