File:Itnem20plot.jpg

Iterates of the Nemtsov function with parameter $q\!=\!2$:

$y=\mathrm{Nem}_2^{\,n}(x)$

versus $x$ for various values of $n$.

C++ generator of curves
typedef std::complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 6) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)
 * 4) include "conto.cin"

DB Q=1.; // Reassigned in the program z_type nem(z_type z){ return z*(1.+z*z*(1.+z*Q)); } z_type nem1(z_type z){ return 1.+z*z*(3.+z*(4.*Q)); }  // WARNING: Q is global!

DB K=Q*Q; // Reassigned in the program DB A[37][19]; int co{ return 0;}
 * 1) include "sunema.txt"

z_type sun(z_type z){ z_type p[37],s; int m,M,n,N; z_type L=log(-z); z_type t=1./sqrt(-2.*z); M=36; for(m=2;m0;n--) { s+=A[m][n]; s*=L; } p[m]=A[m][0]+s; } s=p[M]*t; for(m=M-1; m>1;m--) { s+=p[m]; s*=t;} return t*( 1. + t* (-Q + s) ); }

z_type sune(z_type z){ int m,n; DB x,y; z_type s; DB R=20.+20.*Q*Q; x=Re(z); if(x<-R) return sun(z); y=Im(z); if(fabs(y)>R) return sun(z); n=int(x+R); s=sun(z-(0.+n)); DO(m,n) s=nem(s); return s; }

DB AUNE1=0;

z_type sunem(z_type z){ int m,n; DB x,y; z_type s; DB R=20.+20.*Q*Q; x=Re(z); if(x<-R) return sun(z); y=Im(z); if(fabs(y)>R) return sun(z); n=int(x+R); s=sun(z-(n-AUNE1)); DO(m,n) s=nem(s); return s; }

z_type NemZo=nembra(Q); z_type ANemZo=nem(NemZo); DB tr=Re(ANemZo); DB ti=Im(ANemZo);
 * 1) include"nembran.cin"


 * 1) include "arqnem.cin"

DB C[41]; //DB K=Q*Q; int coa{ //C[1]=(0.5 - K)*Q; C[2]=-0.625 + (-0.25 + K/2.)*K; .. return 40; }
 * 1) include "aunemco.txt"

z_type aune0(z_type z){ z_type s; int n; s=C[40]*z; for(n=39;n>0;n--){s+=C[n];s*=z;} return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ; }

z_type aune1(z_type z){ int n, N=20; DO(n,N) z=arqnem(z); return aune0(z)+(0.+n);}

z_type aunem(z_type z){ int n, N=20; DO(n,N) z=arqnem(z); return aune0(z)+(n-AUNE1);}

z_type aune02(z_type z){ z_type s; int n; s=(C[2]*z+C[1])*z; return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) + s ; } z_type aune01(z_type z){ z_type s; int n; s=C[1]*z; return (-.5/z+Q)/z + (1.5+K)*log(z) + (.5*K + (.75+.5*K)*log(2.)) +s ; }

int main{ int Max; int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;

//FILE *o;o=fopen("aunemplo.eps","w");ado(o,4008,4008); //FILE *o;o=fopen("20.eps","w");ado(o,4008,4008); FILE *o;o=fopen("itnem20plo.eps","w");ado(o,4008,4008); fprintf(o,"4 4 translate\n 1000 1000 scale 2 setlinecap\n"); for(m=0;m<41;m+=10){ M(.1*m,0)L(.1*m,4)} for(n=0;n<41;n+=10){ M(0,.1*n)L(4,.1*n)} fprintf(o,".003 W 0 0 0 RGB 2 setlinecap S\n");

// M(-1,ti)L(1,ti) M(-1,-ti)L(1,-ti) M(tr,-1)L(tr,1) fprintf(o,".002 W 0 0 0 RGB 2 setlinecap S\n"); //M(-1,0) L(0,0) M(tr,ti)L(0,0)L(tr,-ti) //fprintf(o,".006 W 1 1 0 RGB 0 setlinecap S\n");

//     Q=1.e-5; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co; coa; AUNE1=Re(aune1(1.)); Q=2.; K=Q*Q; NemZo=nembra(Q); ANemZo=nem(NemZo); tr=Re(ANemZo); ti=Im(ANemZo); co; coa; AUNE1=Re(aune1(1.));

fprintf(o,"1 setlinejoin 1 setlinecap S\n");

DO(m,201){x=.01+.02*m;y=Re(nem(nem(nem(nem(x)))));if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; } DO(m,201){x=.01+.02*m;y=Re(nem(nem(nem(x))));  if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y);  if(y>4) break; } DO(m,201){x=.01+.02*m;y=Re(nem(nem(x)));       if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y);  if(y>4) break; } DO(m,201){x=.01+.02*m;y=Re(nem(x));            if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y);  if(y>4) break; } fprintf(o,".02 W 0 .7 1 RGB S\n");

DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(arqnem(arqnem(x)))));if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; } DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(arqnem(x))));       if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; } DO(m,201){ x=.01+.02*m; y=Re(arqnem(arqnem(x)));               if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; } DO(m,201){ x=.01+.02*m; y=Re(arqnem(x));                       if(m==0) M(x,y) else L(x,y); printf("%9.2lf %9.2lf\n",x,y); if(y>4) break; } fprintf(o,".02 W 1 .5 0 RGB S\n");

M(0,0)L(4.1,4.1) fprintf(o,".02 W .1 1 .1 RGB S\n"); for(n=-20;n<21;n++){ DO(m,200){ x=.02+.02*m; y=Re(sunem(.1*n+aunem(x))); if(m==0) M(x,y) else L(x,y); /* printf("%9.2lf %9.2lf\n",x,y);*/ if(y>4) break; } }

fprintf(o,".004 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf itnem20plo.eps"); system(   "open itnem20plo.pdf"); //mac return 0;}

Latex generator of labels
\documentclass[12pt]{article} \usepackage{graphics} \paperwidth 4020pt \paperheight 4020pt \usepackage{geometry} \usepackage{rotating} \textwidth 4400pt \textheight 4400pt \topmargin -97pt \oddsidemargin -74pt \parindent 0pt \pagestyle{empty} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \begin{document} \begin{picture}(4008,4008) %\put(20,3840){\sx{20}{$y\!=\!\mathrm{AuNem}_q(x)$}} %\put(20,3840){\sx{20}{$y\!=\!\mathrm{nem}_2^n(x)$}} \put(20,3900){\sx{20}{$y$}} \put(20,2944){\sx{20}{$3$}} \put(20,1940){\sx{20}{$2$}} \put(20, 940){\sx{20}{$1$}} %\put(10, 936){\sx{20}{$-1$}} \put( 946,30){\sx{20}{$1$}} \put(1950,30){\sx{20}{$2$}} \put(2954,30){\sx{20}{$3$}} \put(3870,30){\sx{20}{$x$}} % %\put(0,0){\ing{aunemplo}} \put(0,0){\ing{itnem20plo}} % \put( 340,3450){\sx{16}{\rot{89} $n\!=\!4$\ero}} \put( 482,3450){\sx{16}{\rot{88} $n\!=\!3$\ero}} \put( 620,3450){\sx{16}{\rot{87} $n\!=\!2$\ero}} \put(1010,3480){\sx{16}{\rot{84} $n\!=\!1$\ero}} \put(1578,3420){\sx{16}{\rot{76} $n\!=\!0.5$\ero}} \put(1786,3420){\sx{16}{\rot{73} $n\!=\!0.4$\ero}} \put(2040,3420){\sx{16}{\rot{66} $n\!=\!0.3$\ero}} \put(2390,3420){\sx{16}{\rot{62} $n\!=\!0.2$\ero}} \put(2730,3250){\sx{16}{\rot{54} $n\!=\!0.1$\ero}} \put(3100,3020){\sx{18}{\rot{45} $n\!=\!0$\ero}} \put(3190,2560){\sx{16}{\rot{35} $n\!=\!-0.1$\ero}} \put(3280,2200){\sx{16}{\rot{28} $n\!=\!-0.2$\ero}} \put(3336,1906){\sx{16}{\rot{21} $n\!=\!-0.3$\ero}} \put(3350,1650){\sx{16}{\rot{17} $n\!=\!-0.4$\ero}} \put(3360,1460){\sx{16}{\rot{13} $n\!=\!-0.5$\ero}} \put(3480,908){\sx{16}{\rot{4} $n\!=\!-1$\ero}} \put(3480,518){\sx{16}{\rot{1} $n\!=\!-2$\ero}} \put(3480,382){\sx{16}{\rot{0} $n\!=\!-3$\ero}} \put(3480,238){\sx{16}{\rot{0} $n\!=\!-4$\ero}} % \put( 840,2020){\sx{16}{\rot{83} $y\!=\!\mathrm{Nem}_2(x)$\ero}} \put(1570,1500){\sx{16}{\rot{45} $y\!=\!x$\ero}} \put(1800,720){\sx{16}{\rot{7} $y\!=\!\mathrm{ArqNem}_2(x)$\ero}} \end{picture} \end{document}