Difference between revisions of "Mandelbrot polynomial"

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(Created page with "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. ...")
 
(SuperMandelbrot)
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Then, following the general rule, through the [[superfunction]] $\Phi$ and the [[Abel Function]] $\Psi$,
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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:
 
the $n$th [[iteration]] of the [[Mandelbrot operator]] can be written as follows:
 
$P_c^n(z)=\Phi(n+\Psi(z))$
 
$P_c^n(z)=\Phi(n+\Psi(z))$
   
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.
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In this expression, number $n$ of iterations has no need to be integer; the [[Mandelbrot operator]] can be iterated arbitrary (even complex) number of times.
   
 
==References==
 
==References==

Revision as of 19:11, 16 July 2013

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

References

http://en.wikipedia.org/wiki/Mandelbrot_set

Keywords

Iteration , Logistic operator , Mandelbrot set , Superfunction