File:PowIteT.jpg

${\displaystyle n}$

${\displaystyle y=\mathrm {Pow} _{2}^{~n}(z)=T^{n}(x)~}$

${\displaystyle n}$

${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (1)~~~T(z)=\mathrm {Pow} _{2}(z)=z^{2}=\exp {\Big (}\ln(z)\,2{\Big )}}$

The iteration of ${\displaystyle T}$ is implemented in the following way:  :

${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (2)~~~y=T^{n}(x)={\mathrm {Pow} _{2}}^{n}(z)=F{\Big (}n+G(x){\Big )}}$

where

${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (3)~~~F(z)=\exp {\Big (}\exp {\big (}\ln(2)\,z{\big )}{\Big )}}$

is superfunction for the transfer function ${\displaystyle T}$ and

${\displaystyle \!\!\!\!\!\!\!\!\!\!\ (4)~~~G(z)=\ln {\Big (}\ln(z){\Big )}/\ln(2)}$

is the Abel Function. The thick lines represent ${\displaystyle y=x^{2}}$and ${\displaystyle y={\sqrt {x}}}$. FIrst of them overlaps with <matm>y=T^1(x)</math> and corresponds to ${\displaystyle n\!=\!1}$, and the second overlaps with ${\displaystyle y=T^{-1}(x)}$ and corresponds to ${\displaystyle n\!=\!-1}$.

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