File:Fracit10t150.jpg

Iterate of linear fraction;

$\displaystyle f(z)=\frac{x}{c+z}$ at $c\!=\!1$.

In general the $n$th iterate of $f$ can be expressed as follows:

$\displaystyle f^n(z)=\frac{z}{c^n+\frac{1-c^n}{1-c} z}$

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

For $c=1$, the limit should be considered.

Generator of curves
// File ado.cin should be loaded to 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 c=1.;

//DB F(DB n,DB x){ DB cn=pow(c,n); DB r=(1.-cn)/(1.-c); return x/( cn + r*x); } DB F(DB n,DB x){ if(c==1.) return x/(1.+n*x); DB cn=pow(c,n); DB r=(1.-cn)/(1.-c); return x/( cn + r*x); }

main{ FILE *o; int m,n,k; DB x,y,t; o=fopen("fracit10.eps","w"); ado(o,702,702);
 * 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);

fprintf(o,"101 101 translate 100 100 scale 2 setlinecap\n"); for(n=-1;n<7;n++) { M(-1,n)L(6,n)} for(m=-1;m<7;m++) { M(m,-1)L(m,6)} fprintf(o,".01 W S\n");

n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F(-4.,x);if(y>-10.4&&y<10.4){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 1 0 1 RGB S\n"); n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F(-3.,x);if(y>-8.4&&y<8.4){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 1 0 1 RGB S\n"); n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F(-2.,x);if(y>-7.4&&y<7.4){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 1 0 1 RGB S\n"); n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F(-1.,x);if(y>-7.4&&y<7.4){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 1 0 1 RGB S\n"); n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F( 1.,x);if(y>-7.4&&y<7.4){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 0 1 0 RGB S\n"); n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F( 2.,x);if(y>-7.4&&y<7.4){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 0 1 0 RGB S\n"); n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F( 3.,x);if(y>-8.4&&y<8.4){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 0 1 0 RGB S\n"); n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F( 4.,x);if(y>-10.4&&y<10.4){if(n==0){M(x,y)n=1;}else L(x,y)} else n=0;} fprintf(o,".03 W 0 1 0 RGB S\n");

DO(k,41){ t=-2.+.1*k; n=0;DO(m,3501){x=-1.+.002*(m-.5);y=F(t,x);if(y>-7.2&&y<7.2){ if(n==0){M(x,y) n=1;}else L(x,y)} else n=0;} fprintf(o,".01 W 0 0 0 RGB S\n"); }

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf fracit10.eps"); system(   "open fracit10.pdf"); } //

Latex generator of labels
%File Fracit20t.pdf should be generated with the code above in order to compile the Latex document below.

% \documentclass[12pt]{article} \paperwidth 706pt \paperheight 706pt \textwidth 800pt \textheight 800pt \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}%H0H1H2HHHHHHHHHHHHHH \begin{picture}(704,704)

\put(79,684){\sx{3}{$y$}} \put(79,592){\sx{3}{$5$}} \put(79,492){\sx{3}{$4$}} \put(79,392){\sx{3}{$3$}} \put(79,292){\sx{3}{$2$}} \put(79,192){\sx{3}{$1$}} \put(79,92){\sx{3}{$0$}} \put(94,74){\sx{3}{$0$}} \put(194,74){\sx{3}{$1$}} \put(294,74){\sx{3}{$2$}} \put(394,74){\sx{3}{$3$}} \put(494,74){\sx{3}{$4$}} \put(594,74){\sx{3}{$5$}} \put(686,75){\sx{3}{$x$}} %\put(0,0){\ing{fracit05}} %\put(0,0){\ing{fracit10}} \put(0,0){\ing{fracit10}}

\put(20,200){\rot{70}\sx{3.2}{$n\!=\!2$}\ero} \put(40,154){\rot{69}\sx{3.2}{$n\!=\!3$}\ero} \put(6,122){\rot{22}\sx{3.2}{$n\!=\!4$}\ero}

\put(128, 0){\rot{49}\sx{2.9}{$n\!=\!-4$}\ero} \put(203, 43){\rot{4}\sx{2.9}{$n\!=\!-3$}\ero} \put(207,6){\rot{11}\sx{2.9}{$n\!=\!-2$}\ero}

%\put(139,560){\rot{89}\sx{3.2}{$n\!=\!-2$}\ero} \put(194,560){\rot{87}\sx{3.2}{$n\!=\!-1$}\ero} \put(252,558){\rot{85}\sx{3}{$n\!=\!-0.5$}\ero} \put(274,558){\rot{83}\sx{3}{$n\!=\!-0.4$}\ero} \put(304,558){\rot{79}\sx{3}{$n\!=\!-0.3$}\ero} \put(352,558){\rot{76}\sx{3}{$n\!=\!-0.2$}\ero} \put(424,558){\rot{64}\sx{3}{$n\!=\!-0.1$}\ero}

\put(580,567){\rot{45}\sx{3}{$n\!=\!0$}\ero} \put(610,429){\rot{23}\sx{3}{$n\!=\!0.1$}\ero} \put(608,343){\rot{14}\sx{3}{$n\!=\!0.2$}\ero} \put(607,294){\rot{8}\sx{3}{$n\!=\!0.3$}\ero} \put(606,260){\rot{6}\sx{3}{$n\!=\!0.4$}\ero} \put(605,236){\rot{4}\sx{3}{$n\!=\!0.5$}\ero} \put(620,178){\sx{3.2}{$n\!=\!1$}} \put(620,139){\sx{3.2}{$n\!=\!2$}} \end{picture} \end{document} %