File:IterPowPlotT.png

Explicit plot of $c$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:

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

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

#include<math.h>
#include<stdio.h>
#include<stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#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("IterPowPlot.eps","w"); ado(o,1010,1010);
fprintf(o,"1 1 translate 100 100 scale\n");
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);
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,42){x=0.001+.1*m; y=exp(2.*log(x)); if(m==0)M(x,y) else L(x,y);} fprintf(o,"1 setlinecap 1  setlinejoin .04 W 0 1 0 RGB S\n");
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");
for(n=-12;n<13;n++){t=.1*n; DO(m,92){x=1+.1*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 IterPowPlot.eps");    
      system(    "open IterPowPlot.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
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