File:PowIteT.jpg

Explicit plot of $n$th Iteration of the quadratic function, id est, power function for power 2; $y=\mathrm{Pow}_2^{~c}(z)=T^c(x)~$ for various values of number $c$ of iteration. Here,


 * $\!\!\!\!\!\!\!\!\!\!\ (1) ~ ~ ~ T(z)=\mathrm{Pow}_2(z)=z^2=\exp\Big(\ln(z)\,2\Big)$

The iteration of $T$ is implemented in the following way:


 * $\!\!\!\!\!\!\!\!\!\!\ (2) ~ ~ ~y=T^n(x)={\mathrm{Pow}_2}^n(z)=F\Big(n+G(x)\Big)$

where
 * $\!\!\!\!\!\!\!\!\!\!\ (3) ~ ~ ~ F(z)=\exp\Big( \exp\big( \ln(2)\, z\big)\Big)$

is superfunction for the transfer function $T$ and
 * $\!\!\!\!\!\!\!\!\!\!\ (4) ~ ~ ~G(z)=\ln\Big( \ln(z) \Big)/ \ln(2)$

is the Abel function.

The thick lines represent $y=x^2$ and $y=\sqrt{x}$. FIrst of them overlaps with $y=T^1(x)$ and corresponds to $c\!=\!1$, and the second overlaps with $y=T^{-1}(x)$ and corresponds to $c\!=\!-1$.

Warning
For
 * $\!\!\!\!\!\!\!\!\!\!\ (5) ~ ~ ~T(z)=\mathrm{Pow}_b(z)=z^b$

the iterate can be expressed in the closed form through the same function. For other functions, such a representation may be not available.

C++ Generator of curves
// File ado.cin should be loaded in the working directory in order to compile the code below.

using namespace std; typedef complex z_type;
 * 1) include
 * 2) include
 * 3) include
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "ado.cin"

DB B=2.; DB F(DB z) { return exp( exp( log(B)*z));} DB G(DB z) { return log( log(z) )/log(B);}

main{ int m,n; double x,y,t; FILE *o; o=fopen("PowIte.eps","w"); ado(o,1010,1010); fprintf(o,"1 1 translate 100 100 scale\n"); for(m=0;m<11;m++) {M(m,0)L(m,10)} for(m=0;m<11;m++) {M(0,m)L(10,m)} fprintf(o,"2 setlinecap .01 W S\n"); DO(m,402){x=0.001+.01*m; y=exp(2.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;} DO(m,402){x=0.001+.01*m; y=exp(4.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;} DO(m,402){x=0.001+.01*m; y=exp(8.*log(x)); if(m==0)M(x,y) else L(x,y); if(y>10.1) break;} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 0 1 0 RGB S\n");
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

DO(m,1002){x=0.001+.01*m; y=exp(.5*log(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n"); DO(m,1002){x=0.001+.01*m; y=exp(.25*log(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n"); DO(m,1002){x=0.001+.01*m; y=exp(.125*log(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1 setlinejoin .04 W 1 0 1 RGB S\n"); for(n=-20;n<21;n++){t=.1*n; DO(m,182){x=1.+.05*m; y=F(t+G(x)); if(m==0) M(x,y) else L(x,y); if(y>10.1)break;} } fprintf(o,"1 setlinecap 1 setlinejoin .02 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf PowIte.eps"); system(   "open PowIte.pdf"); getchar; system("killall Preview"); }

Latex Generator of labels
% % file IterPowPlot.pdf should be generated with the code above in order to compile the Latex document below. % % Copyleft 2012 by Dmitrii Kouznetsov % \documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphicx} % \usepackage{rotating} % \paperwidth 1008pt % \paperheight 1008pt % \topmargin -94pt % \oddsidemargin -81pt % \textwidth 1100pt % \textheight 1100pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \parindent 0pt% \pagestyle{empty} % \begin{document} % \begin{picture}(1002,1002) % %\put(10,10){\ing{IterPowPlot}} % \put(10,10){\ing{PowIte}} % %\put(11,976){\sx{4}{$y\!=\!\mathrm{pow}_2^c(x)$}} % \put(11,976){\sx{4}{$y\!=\!x^{2^n}$}} % \put(11,898){\sx{4}{$9$}} % \put(11,798){\sx{4}{$8$}} % \put(11,698){\sx{4}{$7$}} % \put(11,598){\sx{4}{$6$}} % \put(11,498){\sx{4}{$5$}} % \put(11,398){\sx{4}{$4$}} % \put(11,298){\sx{4}{$3$}} % \put(11,198){\sx{4}{$2$}} % \put(11,098){\sx{4}{$1$}} % % \put(100,16){\sx{4}{$1$}} % \put(200,16){\sx{4}{$2$}} % \put(301,16){\sx{4}{$3$}} % \put(401,16){\sx{4}{$4$}} % \put(502,16){\sx{4}{$5$}} % \put(602,16){\sx{4}{$6$}} % \put(703,16){\sx{4}{$7$}} % \put(803,16){\sx{4}{$8$}} % \put(903,16){\sx{4}{$9$}} % \put(990,16){\sx{4}{$x$}} % % \put(140,870){\sx{3.6}{\rot{88}$n\!=\!3$\ero}} % \put(179,870){\sx{3.6}{\rot{87}$n\!=\!2$\ero}} % \put(325,921){\sx{3.6}{\rot{81}$n\!=\!1$\ero}} % \put(461,893){\sx{3.6}{\rot{70}$n\!=\!0.6$\ero}} % \put(509,894){\sx{3.6}{\rot{69}$n\!=\!0.5$\ero}} % \put(563,895){\sx{3.6}{\rot{63}$n\!=\!0.4$\ero}} % \put(630,897){\sx{3.6}{\rot{59}$n\!=\!0.3$\ero}} % \put(713,900){\sx{3.6}{\rot{54}$n\!=\!0.2$\ero}} % \put(826,913){\sx{3.6}{\rot{50}$n\!=\!0.1$\ero}} % % \put(928,932){\sx{3.6}{\rot{45}$n\!=\!0$\ero}} % % \put(894,778){\sx{3.6}{\rot{39}$n\!=\!-0.1$\ero}} % \put(888,676){\sx{3.6}{\rot{33}$n\!=\!-0.2$\ero}} % \put(884,596){\sx{3.6}{\rot{28}$n\!=\!-0.3$\ero}} % \put(877,528){\sx{3.6}{\rot{24}$n\!=\!-0.4$\ero}} % \put(871,476){\sx{3.6}{\rot{19}$n\!=\!-0.5$\ero}} % \put(870,429){\sx{3.6}{\rot{17}$n\!=\!-0.6$\ero}} % % \put(890,298){\sx{3.6}{\rot{8}$n\!=\!-1$\ero}} % \put(886,157){\sx{3.6}{\rot{1}$n\!=\!-2$\ero}} % \put(886,115){\sx{3.6}{\rot{.3}$n\!=\!-3$\ero}} % \end{picture} % \end{document} % %