File:Sqrt2diimap80.jpg

Complex map of iterate number i of exponent to base $\sqrt{2}$ constructed at its lower ("down") fixed point 2:

$u\!+\!\mathrm i v= \exp_{\sqrt{2},\mathrm d}(x\!+\!\mathrm i y)$

Usage: this is figure 16.11 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015. (Numeration of figures in the English version may be different from that of the Russian version.)

The algorithm of the evaluation is also described in the article . (top right map)

C++ generator of the map
Files ado.cin, conto.cin, sqrt2f21e.cin sqrt2f21l.cin should be loaded in order to compile the code below.

// using namespace std; 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++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) include "conto.cin"


 * 1) include "sqrt2f45e.cin"
 * 2) include "sqrt2f45l.cin"
 * 3) include "sqrt2f21e.cin"
 * 4) include "sqrt2f21l.cin"

int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; int M=801,M1=M+1; int N=405,N1=N+1; DB X[M1],Y[N1], g[M1*N1],f[M1*N1], w[M1*N1]; // w is working array. char v[M1*N1]; // v is working array // FILE *o;o=fopen("04.eps","w"); ado(o,202,202); FILE *o;o=fopen("sqrt2diima.eps","w"); ado(o,202,202); fprintf(o,"101 101 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-10.+.025*(m-.5); //DO(n,N1) Y[n]=-10.+.04*(n-.5); // DO(n,200) Y[n]=sinh(3.*(n-200.5)/200.); // DO(n,200) Y[n]=-10.+.05*(n-.5); //        Y[200]=-.0001; //        Y[201]= .0001; for(n=0;n-201. && p<201. && q>-201. && q<201.              && fabs(p)>1.e-14               && fabs(q)>1.e-14                ) { g[m*N1+n]=p; f[m*N1+n]=q;} }} p=2; q=.5; for(m=-10;m<10;m++)for(n=2             ;n<10;n+=2)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q,q);  fprintf(o,".014 W 0 .7 0 RGB S\n"); for(m=0;m<11;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);                 fprintf(o,".014 W 1 0 0 RGB S\n"); for(m=0;m<10;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);                   fprintf(o,".014 W 0 0 1 RGB S\n"); for(m= 1;m<40;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".03 W .8 0 0 RGB S\n"); for(m= 1;m<40;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".03 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-2*p,2*p); fprintf(o,".03 W .5 0 .5 RGB S\n"); for(m=-40;m<41;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".03 W 0 0 0 RGB S\n");

// #include "plofu.cin" M(10,0)L(4,0)fprintf(o,"0 setlinecap .036 W 1 1 1 RGB S\n"); for(n=0;n<17;n++){ M(4+.5*(n+.2),0) L(4+.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n"); for(n=0;n<17;n++){ M(4+.5*(n+.7),0) L(4+.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n"); fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf sqrt2diima.eps"); system(   "open sqrt2diima.pdf"); //for macintosh }

Latex generator of labels
Files generated with codes above should be loaded in order to compile the code below.

\documentclass[12pt]{article} \paperwidth 2072px \paperheight 2076px \textwidth 2394px \textheight 2300px \topmargin -97px \oddsidemargin -78px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \parindent 0pt \pagestyle{empty} \begin{document}\parindent 0pt \sx{10}{\begin{picture}(206,206) %\put(6,5){\ing{Esqrt2ite13Map}} %\put(6,5){\ing{04}} \put(6,5){\ing{sqrt2diima}} \put(2,203.4){\sx{.7}{$y$}} \put(2,184){\sx{.6}{$8$}} \put(2,164){\sx{.6}{$6$}} \put(2,144){\sx{.6}{$4$}} \put(2,124){\sx{.6}{$2$}} \put(2,104){\sx{.6}{$0$}} \put(-2.2,84){\sx{.6}{$-2$}} \put(-2.2,64){\sx{.6}{$-4$}} \put(-2.2,44){\sx{.6}{$-6$}} \put(-2.2,24){\sx{.6}{$-8$}} \put(-2,-1){\sx{.7}{$-\!10$}} \put( 22,-1){\sx{.7}{$-8$}} \put( 42,-1){\sx{.7}{$-6$}} \put( 62,-1){\sx{.7}{$-4$}} \put( 82,-1){\sx{.7}{$-2$}} \put(106,-1){\sx{.7}{$0$}} \put(126,-1){\sx{.7}{$2$}} \put(146,-1){\sx{.7}{$4$}} \put(166,-1){\sx{.7}{$6$}} \put(186,-1){\sx{.7}{$8$}} \put(204,-1){\sx{.7}{$x$}} \put(174,103.5){\sx{.99}{\bf cut}} \put(118,172.3){\sx{.99}{\rot{-16}$v\!=\!3$\ero}} %\put(146,135){\sx{.99}{\rot{-29}$v\!=\!2$\ero}} %\put(148,117){\sx{.99}{\rot{-33}$v\!=\!1$\ero}} \put(125,139){\sx{.99}{\rot{-11}$v\!=\!2$\ero}} \put(126,118){\sx{.99}{\rot{-3}$v\!=\!1$\ero}} \put(121,103){\sx{.99}{\rot{6}$v\!=\!0$\ero}} \put(119,94){\sx{.99}{\rot{3}$v\!=\!-1$\ero}} % \put(27,140){\sx{.99}{\rot{-30}$u\!=\!1$\ero}} \put(91,91){\sx{.99}{\rot{7}$u\!=\!0$\ero}} % \put(117,180){\sx{.99}{\rot{74}$u\!=\!2$\ero}} \put(155,168){\sx{.99}{\rot{58}$u\!=\!3$\ero}} \put(155,120){\sx{.99}{\rot{55}$u\!=\!4$\ero}} \end{picture}} \end{document}