Difference between revisions of "File:Logi1a345T300.png"

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{{oq|File:Logi1a345T300.png|Original file ‎(1,636 × 565 pixels, file size: 184 KB, MIME type: image/png) }}
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Importing image file
  
  +
  +
[[Explicit plot]]s of various iterates of the [[Logistic operator]] with various values of parameter \(s\).
  +
  +
\(y=\mathrm{LogisticOperator}_s^{\,c}(x)\) for
  +
\(s\!=\!3\) (left),
  +
\(s\!=\!4\) (center),
  +
\(s\!=\!5\) (right)
  +
at
  +
\(c=1\) (black),
  +
\(c=0.8\) (blue),
  +
\(c=0.5\) (green),
  +
\(c=0.2\) (red).
  +
  +
The non-integer iterates of the [[logisticOperator]] are calculated through the functions [[LogisticSequence]] and [[ArcLogisticSequence]] with
  +
  +
\[ \mathrm{LogisticOperator}_s^{\,c}(z)=
  +
\mathrm{LogisticSequence}_s\Big( c + \mathrm{ArcLogisticSequence}(z) \big)
  +
\]
  +
  +
where [[logisticOperator]] is quadratic function of special kind:
  +
  +
\(\mathrm{LogisticOperator}_s(z)=sz(1\!-\!z)\)
  +
  +
The picture is used as Figure 7.2 at page 72 of book «[[Superfunctions]]»<ref>
  +
https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas,algorithms,tables,graphics - 2020/7/28
  +
</ref><ref>https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). [[Superfunctions]]: Non-integer iterates of holomorphic functions. [[Tetration]] and other [[superfunction]]s. Formulas, algorithms, tables, graphics. Publisher: [[Lambert Academic Publishing]].
  +
</ref>
  +
<br>
  +
in order to show that for some values of the argument, the non-integer iterate is not real.
  +
The cut line appears in the complex map due to the cut of the [[Abelfunction]],
  +
this cut line is marked at many ficures of Chapter 7 of that book.
  +
  +
==C++ generator of curves for \(s=3\) ==
  +
// Files [[efjh.cin]] and [[ado.cin]] should be loaded in order to compile the [[C++]] code below.
  +
<pre>
  +
#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"
  +
#include "efjh.cin"
  +
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
FILE *o;o=fopen("logi1a3.eps","w");ado(o,104,104);
  +
fprintf(o,"2 2 translate\n 100 100 scale\n");
  +
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
  +
M(0,0)L(1,0)L(1,1)L(0,1)
  +
fprintf(o,"C .003 W 0 0 0 RGB S\n");
  +
M(0,.25)L(1,.25) M(.25,0)L(.25,1)
  +
M(0,.50)L(1,.50) M(.50,0)L(.50,1)
  +
M(0,.75)L(1,.75) M(.75,0)L(.75,1)
  +
fprintf(o,".003 W 0 0 0 RGB S\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
maq(3.);
  +
DO(m,101){x=1.-.0000999*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n");
  +
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.8+E(x))); L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n");
  +
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n");
  +
M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.2+E(x))); L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf logi1a3.eps");
  +
system( "open logi1a3.eps");
  +
getchar(); system("killall Preview");
  +
}
  +
</pre>
  +
==C++ generator of curves for \(s=4\) ==
  +
// Files [[efjh.cin]] and [[ado.cin]] should be loaded in order to compile the [[C++]] code below.
  +
<pre>
  +
#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"
  +
#include "efjh.cin"
  +
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
FILE *o;o=fopen("logi1a4.eps","w");ado(o,104,104);
  +
fprintf(o,"2 2 translate\n 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);
  +
M(0,0)L(1,0)L(1,1)L(0,1)
  +
fprintf(o,"C .003 W 0 0 0 RGB S\n");
  +
M(0,.25)L(1,.25) M(.25,0)L(.25,1)
  +
M(0,.50)L(1,.50) M(.50,0)L(.50,1)
  +
M(0,.75)L(1,.75) M(.75,0)L(.75,1)
  +
fprintf(o,".003 W 0 0 0 RGB S\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
maq(4.);
  +
DO(m,101){x=1.-.0000999*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n");
  +
M(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.8+E(x))); L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n");
  +
M(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n");
  +
M(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.2+E(x))); L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf logi1a4.eps");
  +
system( "open logi1a4.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
</pre>
  +
==C++ generator of curves for \(s=5\) ==
  +
// FIels [[efjh.cin]] and [[ado.cin]] should be loaded to the working directory in order to compile the [[C++]] code below.
  +
//<pre>
  +
#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"
  +
#include "efjh.cin"
  +
  +
main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  +
FILE *o;o=fopen("logi1a5.eps","w");ado(o,140,140);
  +
fprintf(o,"2 2 translate\n 100 100 scale\n");
  +
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
  +
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
  +
M(0,0)L(1.25,0)L(1.25,1.25)L(0,1.25)
  +
fprintf(o,"C .003 W 0 0 0 RGB S\n");
  +
M(0,.25)L(1.25,.25) M(.25,0)L(.25,1.25)
  +
M(0,.50)L(1.25,.50) M(.50,0)L(.50,1.25)
  +
M(0,.75)L(1.25,.75) M(.75,0)L(.75,1.25)
  +
M(0,1.0)L(1.25,1.0) M(1.0,0)L(1.0,1.25)
  +
fprintf(o,".003 W 0 0 0 RGB S\n");
  +
fprintf(o,"1 setlinejoin 2 setlinecap\n");
  +
maq(5);
  +
DB x0=1.25;
  +
DO(m,101){x=1.001-.00010001*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n");
  +
DO(m,101){x=1.17-.000116*m*m;y=Re(F(.8+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n");
  +
M(1.25,-.02)
  +
DO(m,101){x=x0-.0001249*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n");
  +
M(1.25,-.01)
  +
DO(m,101){x=x0-.0001249*m*m;y=Re(F(.2+E(x))); L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n");
  +
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
  +
system("epstopdf logi1a5.eps");
  +
system( "open logi1a5.pdf");
  +
getchar(); system("killall Preview");
  +
}
  +
</pre>
  +
  +
==Latex generator of labels==
  +
  +
% Files logi1a3.pdf, logi1a4.pdf, logi1a5.pdf should be generated with the codes above in order to compile the [[Latex]] document below.
  +
%<pre>
  +
\documentclass[12pt]{article} %<br>
  +
\usepackage{geometry} %<br>
  +
\usepackage{graphics} %<br>
  +
\usepackage{rotating} %<br>
  +
\paperwidth 394pt %<br>
  +
\paperheight 136pt %<br>
  +
\topmargin -100pt %<br>
  +
\oddsidemargin -75pt %<br>
  +
\newcommand \sx {\scalebox} %<br>
  +
\newcommand \ing \includegraphics %<br>
  +
\newcommand \rot {\begin{rotate}} %<br>
  +
\newcommand \ero {\end{rotate}} %<br>
  +
\parindent 0pt %<br>
  +
\pagestyle{empty} %<br>
  +
\begin{document} %<br>
  +
\newcommand \fiax { %<br>
  +
\put(3,108){\sx{.9}{1}} %<br>
  +
%\put(-9, 84){\sx{.9}{0.25}} %<br>
  +
\put(3, 58){\sx{.9}{$\frac{1}{2}$}} %<br>
  +
%\put(-9, 34){\sx{.9}{0.25}} %<br>
  +
\put(3, 8){\sx{.9}{0}} %<br>
  +
\put( 8, 1){\sx{.9}{0}} %<br>
  +
\put( 55, 1){\sx{.9}{0.5}} %<br>
  +
\put(108, 1){\sx{.9}{1}} %<br>
  +
} %<br>
  +
\begin{picture}(126,124) %<br>
  +
\put(8,8){\ing{logi1a3}} \fiax \put(2,124){\sx{1}{$y\!=\!T^{c}(x)$}} %<br>
  +
\put(31,63){\rot{51}\sx{.85}{$c\!=1$}\ero} %<br>
  +
\put(33,53){\rot{49}\sx{.85}{$c\!=0.8$}\ero} %<br>
  +
\put(39,48){\rot{48}\sx{.85}{$c\!=0.5$}\ero} %<br>
  +
\put(44,40){\rot{46}\sx{.85}{$c\!=0.2$}\ero} %<br>
  +
\put(118, 1){\sx{.9}{$x$}} %<br>
  +
\put(44,20){\sx{1.6}{$s\!=\!3$}} %<br>
  +
\end{picture} %<br>
  +
\begin{picture}(126,124) \put(8,8){\ing{logi1a4}} \fiax \put(2,124){\sx{1}{$y\!=\!T^{c}(x)$}} %<br>
  +
\put(32,84){\rot{57}\sx{.9}{$c\!=1$}\ero} %<br>
  +
\put(39,70){\rot{54}\sx{.88}{$c\!=0.8$}\ero} %<br>
  +
\put(46,65){\rot{51}\sx{.88}{$c\!=0.5$}\ero} %<br>
  +
\put(54,54){\rot{49}\sx{.9}{$c\!=0.2$}\ero} %<br>
  +
\put(118, 1){\sx{.9}{$x$}} %<br>
  +
\put(44,20){\sx{1.6}{$s\!=\!4$}} %<br>
  +
\end{picture} %<br>
  +
\begin{picture}(140,127) \put(8,8){\ing{logi1a5}} \fiax %\put(0,124){\sx{1}{$y$}} %<br>
  +
\put(34,107){\rot{61}\sx{.93}{$c\!=1$}\ero} %<br>
  +
\put(45, 90){\rot{59}\sx{.92}{$c\!=0.8$}\ero} %<br>
  +
\put(53, 77){\rot{55}\sx{.92}{$c\!=0.5$}\ero} %<br>
  +
\put(58, 60){\rot{50}\sx{.92}{$c\!=0.2$}\ero} %<br>
  +
\put(128, 1){\sx{.9}{$x$}} %<br>
  +
\put(44,20){\sx{1.6}{$s\!=\!5$}} %<br>
  +
\end{picture} %<br>
  +
\end{document} %<br>
  +
%</pre>
  +
  +
==References==
  +
{{ref}}
  +
  +
{{fer}}
  +
  +
==Keywords==
  +
«[[Holomorphic extension of the Collatz subsequence]]»,
  +
«[[LogisitcOperator]]»,
  +
«[[Table of superfunctions]]»,
  +
«[[Transfer equation]]»,
  +
«[[Superfunctions]]»,
  +
  +
[[Category:Abel function]]
  +
[[Category:Abelfunction]]
  +
[[Category:ArcLogisticSequence]]
  +
[[Category:Book]]
  +
[[Category:BookPlot]]
  +
[[Category:C++]]
  +
[[Category:Elutin Pavel Vyacheslavocish]]
  +
[[Category:Explicit plot]]
  +
[[Category:Latex]]
  +
[[Category:LogisitcOperator]]
  +
[[Category:Iteration]]
  +
[[Category:LogisticSequence]]
  +
[[Category:Superfunction]]
  +
[[Category:Superfunctions]]
  +
[[Category:Transfer function]]

Latest revision as of 11:44, 21 August 2025


Explicit plots of various iterates of the Logistic operator with various values of parameter \(s\).

\(y=\mathrm{LogisticOperator}_s^{\,c}(x)\) for \(s\!=\!3\) (left), \(s\!=\!4\) (center), \(s\!=\!5\) (right) at \(c=1\) (black), \(c=0.8\) (blue), \(c=0.5\) (green), \(c=0.2\) (red).

The non-integer iterates of the logisticOperator are calculated through the functions LogisticSequence and ArcLogisticSequence with

\[ \mathrm{LogisticOperator}_s^{\,c}(z)= \mathrm{LogisticSequence}_s\Big( c + \mathrm{ArcLogisticSequence}(z) \big) \]

where logisticOperator is quadratic function of special kind:

\(\mathrm{LogisticOperator}_s(z)=sz(1\!-\!z)\)

The picture is used as Figure 7.2 at page 72 of book «Superfunctions»[1][2]
in order to show that for some values of the argument, the non-integer iterate is not real. The cut line appears in the complex map due to the cut of the Abelfunction, this cut line is marked at many ficures of Chapter 7 of that book.

C++ generator of curves for \(s=3\)

// Files efjh.cin and ado.cin should be loaded in order to compile the C++ code below. 

 #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"
 #include "efjh.cin"

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 FILE *o;o=fopen("logi1a3.eps","w");ado(o,104,104);
 fprintf(o,"2 2 translate\n 100 100 scale\n");
 #define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 M(0,0)L(1,0)L(1,1)L(0,1)
 fprintf(o,"C .003 W 0 0 0 RGB S\n");
 M(0,.25)L(1,.25) M(.25,0)L(.25,1)
 M(0,.50)L(1,.50) M(.50,0)L(.50,1)
 M(0,.75)L(1,.75) M(.75,0)L(.75,1)
 fprintf(o,".003 W 0 0 0 RGB S\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 maq(3.);
 DO(m,101){x=1.-.0000999*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n"); 
 M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.8+E(x))); L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n"); 
 M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.5+E(x)));  L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n"); 
 M(.75,9./16.) DO(m,101){x=.75-.0000749*m*m;y=Re(F(.2+E(x)));  L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n"); 
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi1a3.eps");
        system(    "open logi1a3.eps");
        getchar(); system("killall Preview");
 }

C++ generator of curves for \(s=4\)

// Files efjh.cin and ado.cin should be loaded in order to compile the C++ code below. 

 #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"
 #include "efjh.cin"

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 FILE *o;o=fopen("logi1a4.eps","w");ado(o,104,104);
 fprintf(o,"2 2 translate\n 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);
 M(0,0)L(1,0)L(1,1)L(0,1)
 fprintf(o,"C .003 W 0 0 0 RGB S\n");
 M(0,.25)L(1,.25) M(.25,0)L(.25,1)
 M(0,.50)L(1,.50) M(.50,0)L(.50,1)
 M(0,.75)L(1,.75) M(.75,0)L(.75,1)
 fprintf(o,".003 W 0 0 0 RGB S\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 maq(4.);
 DO(m,101){x=1.-.0000999*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n"); 
 M(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.8+E(x))); L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n"); 
 M(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.5+E(x)));  L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n"); 
 M(1,0) DO(m,101){x=1.-.0000999*m*m;y=Re(F(.2+E(x)));  L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n"); 
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi1a4.eps");
        system(    "open logi1a4.pdf");
        getchar(); system("killall Preview");
 }

C++ generator of curves for \(s=5\)

// FIels efjh.cin and ado.cin should be loaded to the working directory in order to compile the C++ code below.

//
 #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"
 #include "efjh.cin"

 main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
 FILE *o;o=fopen("logi1a5.eps","w");ado(o,140,140);
 fprintf(o,"2 2 translate\n 100 100 scale\n");
 #define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 M(0,0)L(1.25,0)L(1.25,1.25)L(0,1.25)
 fprintf(o,"C .003 W 0 0 0 RGB S\n");
 M(0,.25)L(1.25,.25) M(.25,0)L(.25,1.25)
 M(0,.50)L(1.25,.50) M(.50,0)L(.50,1.25)
 M(0,.75)L(1.25,.75) M(.75,0)L(.75,1.25)
 M(0,1.0)L(1.25,1.0) M(1.0,0)L(1.0,1.25)
 fprintf(o,".003 W 0 0 0 RGB S\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 maq(5);
 DB x0=1.25;
 DO(m,101){x=1.001-.00010001*m*m;y=Re(F(1.+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 0 RGB S\n"); 
 DO(m,101){x=1.17-.000116*m*m;y=Re(F(.8+E(x)));if(m==0)M(x,y)else L(x,y);}fprintf(o,".006 W 0 0 .8 RGB S\n"); 
 M(1.25,-.02)
 DO(m,101){x=x0-.0001249*m*m;y=Re(F(.5+E(x))); L(x,y);}fprintf(o,".01 W 0 .8 0 RGB S\n"); 
 M(1.25,-.01)
 DO(m,101){x=x0-.0001249*m*m;y=Re(F(.2+E(x))); L(x,y);}fprintf(o,".006 W .8 0 0 RGB S\n"); 
 fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf logi1a5.eps");
        system(    "open logi1a5.pdf");
        getchar(); system("killall Preview");
 }

Latex generator of labels

% Files logi1a3.pdf, logi1a4.pdf, logi1a5.pdf should be generated with the codes above in order to compile the Latex document below.

%
\documentclass[12pt]{article} %<br>
\usepackage{geometry} %<br>
\usepackage{graphics} %<br>
\usepackage{rotating} %<br>
\paperwidth 394pt %<br>
\paperheight 136pt %<br>
\topmargin -100pt %<br>
\oddsidemargin -75pt %<br>
\newcommand \sx {\scalebox} %<br>
\newcommand \ing \includegraphics %<br>
\newcommand \rot {\begin{rotate}} %<br>
\newcommand \ero {\end{rotate}} %<br>
\parindent 0pt %<br>
\pagestyle{empty} %<br>
\begin{document} %<br>
\newcommand \fiax { %<br>
\put(3,108){\sx{.9}{1}} %<br>
%\put(-9, 84){\sx{.9}{0.25}} %<br>
\put(3, 58){\sx{.9}{$\frac{1}{2}$}} %<br>
%\put(-9, 34){\sx{.9}{0.25}} %<br>
\put(3,  8){\sx{.9}{0}} %<br>
\put(  8,  1){\sx{.9}{0}} %<br>
\put( 55,  1){\sx{.9}{0.5}} %<br>
\put(108, 1){\sx{.9}{1}} %<br>
} %<br>
\begin{picture}(126,124)  %<br>
\put(8,8){\ing{logi1a3}} \fiax \put(2,124){\sx{1}{$y\!=\!T^{c}(x)$}} %<br>
\put(31,63){\rot{51}\sx{.85}{$c\!=1$}\ero} %<br>
\put(33,53){\rot{49}\sx{.85}{$c\!=0.8$}\ero} %<br>
\put(39,48){\rot{48}\sx{.85}{$c\!=0.5$}\ero} %<br>
\put(44,40){\rot{46}\sx{.85}{$c\!=0.2$}\ero} %<br>
\put(118,  1){\sx{.9}{$x$}} %<br>
\put(44,20){\sx{1.6}{$s\!=\!3$}} %<br>
\end{picture} %<br>
\begin{picture}(126,124) \put(8,8){\ing{logi1a4}} \fiax \put(2,124){\sx{1}{$y\!=\!T^{c}(x)$}} %<br>
\put(32,84){\rot{57}\sx{.9}{$c\!=1$}\ero} %<br>
\put(39,70){\rot{54}\sx{.88}{$c\!=0.8$}\ero} %<br>
\put(46,65){\rot{51}\sx{.88}{$c\!=0.5$}\ero} %<br>
\put(54,54){\rot{49}\sx{.9}{$c\!=0.2$}\ero} %<br>
\put(118,  1){\sx{.9}{$x$}} %<br>
\put(44,20){\sx{1.6}{$s\!=\!4$}} %<br>
\end{picture} %<br>
\begin{picture}(140,127) \put(8,8){\ing{logi1a5}} \fiax %\put(0,124){\sx{1}{$y$}} %<br>
\put(34,107){\rot{61}\sx{.93}{$c\!=1$}\ero} %<br>
\put(45, 90){\rot{59}\sx{.92}{$c\!=0.8$}\ero} %<br>
\put(53, 77){\rot{55}\sx{.92}{$c\!=0.5$}\ero} %<br>
\put(58, 60){\rot{50}\sx{.92}{$c\!=0.2$}\ero} %<br>
\put(128,  1){\sx{.9}{$x$}} %<br>
\put(44,20){\sx{1.6}{$s\!=\!5$}} %<br>
\end{picture} %<br>
\end{document} %<br>
%

References

  1. ↑ https://www.amazon.co.jp/Superfunctions-Non-integer-holomorphic-functions-superfunctions/dp/6202672862 Dmitrii Kouznetsov. Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas,algorithms,tables,graphics - 2020/7/28
  2. ↑ https://mizugadro.mydns.jp/BOOK/468.pdf Dmitrii Kouznetsov (2020). Superfunctions: Non-integer iterates of holomorphic functions. Tetration and other superfunctions. Formulas, algorithms, tables, graphics. Publisher: Lambert Academic Publishing.

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