File:Itelin125T.jpg

Iterates of the linear function

$T(z)=A+B z$

for $A\!=\!1$, $B\!=\!2$

$y=T^n(x)$ is plotted versus $x$ for various values of $n$.

C++ Generator of lines
// File ado.cin should be loaded in the working directory in order to compile the C++ code below //
 * 1) include
 * 2) include
 * 3) include
 * 4) define DO(x,y) for(x=0;x<y;x++)
 * 5) define DB double
 * 6) include"ado.cin"

DB A=1.0000; DB B=2.000; DB T(DB c,DB x){ DB Bc=pow(B,c); return A*(Bc-1.)/(B-1.) + Bc*x; } DB U(DB c,DB x){ DB Bc=pow(B,c); return (x-A*(Bc-1.)/(B-1.))/Bc; }

int main{ FILE *o; int m,n,k; DB c, x,y,t; o=fopen("itelin125.eps","w"); ado(o,1002,1002); fprintf(o,"501 501 translate 100 100 scale 2 setlinecap\n"); for(n=-5;n<6;n++) { M(-5,n)L(5,n)} for(m=-5;m<6;m++) { M(m,-5)L(m,5)} fprintf(o,".004 W S\n"); c= 40.001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 0 1 0 RGB S\n"); c= 4.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 0 1 0 RGB S\n"); c= 3.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 0 1 0 RGB S\n"); c= 2.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 0 1 0 RGB S\n"); c= 1.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 0 1 0 RGB S\n"); c= 0.000001; x=-5.;y=T(c,x); if(y<-7.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".02 W 0 0 0 RGB S\n"); c=-1.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 1 0 1 RGB S\n"); c=-2.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 1 0 1 RGB S\n"); c=-3.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 1 0 1 RGB S\n"); c=-4.000001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 1 0 1 RGB S\n"); c=-40.0001; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);fprintf(o,".03 W 1 0 1 RGB S\n"); DO(n,31){c=-3.000001+.2*n; x=-5.;y=T(c,x); if(y<-5.){y=-5.;x=U(c,y);} M(x,y);x=5.;y=T(c,x); if(y>5.){y=5.;x=U(c,y);}L(x,y);} fprintf(o,".012 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf itelin125.eps"); system(   "open itelin125.pdf"); } //
 * 1) define M(x,y) fprintf(o,"%7.4f %7.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%7.4f %7.4f L\n",0.+x,0.+y);

Latex generator of curves
% \documentclass[12pt]{article} \paperwidth 1006pt \paperheight 1006pt \textwidth 1800pt \textheight 1800pt \topmargin -108pt \oddsidemargin -72pt \parindent 0pt \pagestyle{empty} \usepackage {graphics} \usepackage{rotating} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox} \begin{document} \begin{picture}(1004,1004) \put(0,0){\ing{itelin125}} \put(480,984){\sx{3}{$y$}} \put(480,892){\sx{3}{$4$}} \put(480,792){\sx{3}{$3$}} \put(480,692){\sx{3}{$2$}} \put(480,592){\sx{3}{$1$}} \put(479,492){\sx{3}{$0$}} \put(454,391){\sx{3}{$-1$}} \put(454,291){\sx{3}{$-2$}} \put(454,191){\sx{3}{$-3$}} \put(454,91){\sx{3}{$-4$}}

\put(70,475){\sx{3}{$-4$}} \put(170,475){\sx{3}{$-3$}} \put(270,475){\sx{3}{$-2$}} \put(370,475){\sx{3}{$-1$}} \put(495,475){\sx{3}{$0$}} \put(595,475){\sx{3}{$1$}} \put(695,475){\sx{3}{$2$}} \put(795,475){\sx{3}{$3$}} \put(895,475){\sx{3}{$4$}} \put(984,476){\sx{3.1}{$x$}}

\put(410,870){\rot{89}\sx{3.1}{$n\!\rightarrow\!\infty$}\ero} \put(440,870){\rot{84}\sx{3.1}{$n\!=\!4$}\ero} \put(470,870){\rot{82}\sx{3.1}{$n\!=\!3$}\ero} \put(530,870){\rot{74}\sx{3.1}{$n\!=\!2$}\ero} \put(616,870){\rot{65}\sx{3.1}{$n\!=\!1.2$}\ero} \put(646,870){\rot{62}\sx{3.1}{$n\!=\!1$}\ero} \put(682,870){\rot{59}\sx{3.1}{$n\!=\!0.8$}\ero} \put(724,870){\rot{57}\sx{3.1}{$n\!=\!0.6$}\ero} \put(768,870){\rot{53}\sx{3.1}{$n\!=\!0.4$}\ero} \put(822,870){\rot{48}\sx{3.1}{$n\!=\!0.2$}\ero} \put(880,867){\rot{45}\sx{3.1}{$n\!=\!0$}\ero} \put(890,813){\rot{41}\sx{3.1}{$n\!=\!-0.2$}\ero} \put(889,760){\rot{36}\sx{3.1}{$n\!=\!-0.4$}\ero} \put(889,712){\rot{32}\sx{3.1}{$n\!=\!-0.6$}\ero} \put(889,671){\rot{29}\sx{3.1}{$n\!=\!-0.8$}\ero} \put(889,633){\rot{26}\sx{3.1}{$n\!=\!-1$}\ero} \put(905,518){\rot{13}\sx{3.1}{$n\!=\!-2$}\ero} \put(900,422){\rot{4}\sx{3.1}{$n\!=\!-4$}\ero} \put(874,392){\rot{.01}\sx{3.1}{$n\!\rightarrow\!-\infty$}\ero} \end{picture} \end{document}

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