File:Frac1zt.jpg

Iterates of function $T(z)=-1/z$

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

The non-integer iterates of function $T$ are evaluated using the superfunction

$\displaystyle F(z)=\tan\left(\frac{2}{\pi} z\right)$

and the Abel function

$\displaystyle G(z)=F^{-1}(z)=\frac{2}{\pi} \arctan\left( z\right)$

C++ 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-5&&y<5){ 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,1001){x=-5.+.01*(m-.5);y=F(1.,x);if(y>-5&&y<5){ 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,20){ t=-2.+.1*k; n=0;DO(m,1001){x=-5.+.01*(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 frac1z.eps"); system(   "open frac1z.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 labels
% File frac1z.pdf should be generated with the code above in order to compile the Latex document below. \documentclass[12pt]{article} \paperwidth 1006pt \paperheight 1006pt \textwidth 1100pt \textheight 1100pt \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(479,984){\sx{3}{$y$}} \put(479,892){\sx{3}{$4$}} \put(479,792){\sx{3}{$3$}} \put(479,692){\sx{3}{$2$}} \put(479,592){\sx{3}{$1$}} \put(479,492){\sx{3}{$0$}} \put(459,392){\sx{3}{$-1$}} \put(459,292){\sx{3}{$-2$}} \put(459,192){\sx{3}{$-3$}} \put(459, 92){\sx{3}{$-4$}}

\put( 78,474){\sx{3}{$-4$}} \put(178,474){\sx{3}{$-3$}} \put(278,474){\sx{3}{$-2$}} \put(378,474){\sx{3}{$-1$}} \put(494,474){\sx{3}{$0$}} \put(594,474){\sx{3}{$1$}} \put(694,474){\sx{3}{$2$}} \put(794,474){\sx{3}{$3$}} %\put(894,474){\sx{3}{$4$}} \put(986,475){\sx{3}{$x$}} %\put(0,0){\ing{fracit05}} %\put(0,0){\ing{fracit10}} \put(0,0){\ing{frac1z}}

\put( 62,886){\rot{44}\sx{3}{$n\!=\!1.7$}\ero} \put(166,774){\rot{44}\sx{3}{$n\!=\!1.6$}\ero} \put(176,678){\rot{30}\sx{3}{$n\!=\!1.5$}\ero} \put(180,626){\rot{20}\sx{3}{$n\!=\!1.4$}\ero} \put(180,588){\rot{16}\sx{3}{$n\!=\!1.3$}\ero} \put(180,563){\rot{11}\sx{3}{$n\!=\!1.2$}\ero} %\put(180,530){\rot{12}\sx{3}{$n\!=\!1.1$}\ero} \put(180,523){\rot{9}\sx{3}{$n\!=\!1$}\ero} \put(180,441){\rot{7}\sx{3}{$n\!=\!0.5$}\ero} \put(568,866){\rot{84}\sx{3}{$n\!=\!0.5$}\ero} \put(620,866){\rot{80}\sx{3}{$n\!=\!0.3$}\ero} \put(665,866){\rot{78}\sx{3}{$n\!=\!0.2$}\ero} \put(734,866){\rot{69}\sx{3}{$n\!=\!0.1$}\ero} \put(840,827){\rot{45}\sx{3}{$n\!=\!0$}\ero} \put(866,714){\rot{20}\sx{3}{$n\!=\!-0.1$}\ero} \put(866,643){\rot{12}\sx{3}{$n\!=\!-0.2$}\ero} \put(866,602){\rot{8}\sx{3}{$n\!=\!-0.3$}\ero} \put(866,572){\rot{05}\sx{3}{$n\!=\!-0.4$}\ero} \put(866,529){\rot{04}\sx{3}{$n\!=\!-0.6$}\ero} \put(866,466){\rot{02}\sx{3}{$n\!=\!-1$}\ero} \put(866,426){\rot{04}\sx{3}{$n\!=\!-1.2$}\ero} \put(866,368){\rot{07}\sx{3}{$n\!=\!-1.4$}\ero} \put(870,320){\rot{09}\sx{3}{$n\!=\!-1.5$}\ero} \put(872,230){\rot{20}\sx{3}{$n\!=\!-1.6$}\ero} \put(882, 30){\rot{44}\sx{3}{$n\!=\!-1.7$}\ero} \end{picture} \end{document}