File:Esqrt2iterMapT.png

Complex map of 1/3 th iteration of the exponential to base sqrt(2).

$u\!\!\mathrm i v= T^{1/3}(x\!+\!\mathrm i y)$

$T(x) = \Big(\sqrt{2}\Big){^z}= \exp_b(z)$

$b=\sqrt{2}$

The iteration is constructed through the superfunction $F$ constructed with regular iteration at the fixed point $L\!=\!4$, and the Abel function $G=F^{-1}$,

$T^{1/3}(z)=F\!\left(\frac{1}{3}+G(z)\right)$

In publication ^[1], the functions $F$ and $G$ are called $F_{4,5}$ and $F_{4,5}^{~-1}$.

References

C++ generator of curves

// Files ado.cin and conto.cin should be loaded to the working directory in order to compile the C++ code below.

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
using namespace std;
#include <complex>
typedef complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#include "conto.cin"

z_type f45E(z_type z){int n; z_type e,s; 
DB coeu[21]={1.,
0.44858743119526122890, .19037224679780675668, 
0.77829576536968278770e-1, 0.30935860305707997953e-1, 
0.12022125769065893274e-1, 0.45849888965617461424e-2,
0.17207423310577291102e-2, 0.63681090387985537364e-3,
0.23276960030302567773e-3, 0.84145511838119915857e-4,
0.30115646493706434120e-4, 0.10680745813035087964e-4,
0.37565713615564248453e-5, 0.13111367785052622794e-5,
0.45437916254218158081e-6, 0.15642984632975371803e-6,
0.53523276400816416929e-7, 0.18207786280204973113e-7,  
0.61604764947389226744e-8, 0.2e-8};
       e=exp(.32663425997828098238*(z-1.11520724513161));      
       s=coeu[20]; for(n=19;n>=0;n--) { s*=e; s+=coeu[n]; } 
//     s=coeu[19]; for(n=18;n>=0;n--) { s*=e; s+=coeu[n]; } 
       return 4.+s*e;}

z_type F45E(z_type z){ DB b=sqrt(2.);
                       if(Re(z)<-1.) return f45E(z);
                       return exp(F45E(z-1.)*log(b));
               }

z_type f45L(z_type z){ int n; z_type e,s;
DB Uco[21]={1,  
-.44858743119526122890,        .21208912005491969757, 
-.10218436750697167872,        0.49698683037371830337e-1, 
-0.2430759032611955221e-1,     0.11933088396510860210e-1,
-0.587369764200886206e-2,      0.289686728710575713e-2,
-0.1430908106079253664e-2,     0.7076637148565759223e-3,
-0.3503296158729878e-3,        0.17357560046634138e-3,
-0.86061011929162626e-4,       0.426959089012891e-4,
-0.2119302906819844809e-4,     0.1052442259960e-4,
-0.52285174354086e-5,          0.259844999161e-5,
-0.129178211214818578e-5,      0.4e-6  };
       z-=4.;
       s=Uco[19]; for(n=18; n>=0; n--){ s*=z; s+=Uco[n]; }
//     s=Uco[20]; for(n=19; n>=0; n--){ s*=z; s+=Uco[n]; }
//      return log(s*z)/.32663425997828098238 +1.1152091357215375; 
       return log(s*z)/.32663425997828098238 +1.11520724513161;
 }

z_type F45L(z_type z){ DB b=sqrt(2.);
                       if(abs(z-4.)>.4) return F45L(log(z)/log(b))+1. ;
                        return f45L(z);
               }
// #include"sqrt2f45E.cin"

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd;
int M=401,M1=M+1;
int N=403,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("Esqrt2iterMap.eps","w");  ado(o,202,202);
fprintf(o,"101 101 translate\n 10 10 scale\n");
DO(m,M1) X[m]=-10.+.05*(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=202;n<N1;n++) Y[n]=sinh(3.*(n-200.5-2)/200.);
for(n=202;n<N1;n++) Y[n]=-10.+.05*(n-2);
for(m=-10;m<11;m++) {M(m,-10)L(m,10)}
for(n=-10;n<11;n++) {M(  -10,n)L(10,n)} 
fprintf(o,"1 setlinejoin 2 setlinecap\n");
fprintf(o," .006 W 0 0 0 RGB S\n");
// z_type tm,tp,F[M1*N1];
DO(m,M1)DO(n,N1){      g[m*N1+n]=9999;
                       f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; 
DO(n,N1){y=Y[n]; z=z_type(x,y);        
               c=F45L(z); c=F45E(1./3.+c);
               p=Re(c); 
               q=Im(c);
               if(p>-25. && p<25. && q>-25. && q<25.
//              && fabs(p)>1.e-14
//              && fabs(q)>1.e-14
               ) { g[m*N1+n]=p; f[m*N1+n]=q;}
       }}
p=2.5; q=.8;
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,".01 W 0 .6 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,".01 W .9 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,".01 W 0 0 .9 RGB S\n");
for(m= 1;m<25;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<25;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=-24;m<25;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(2,0)fprintf(o,"0 setlinecap .04 W 1 1 1 RGB S\n");
for(n=0;n<21;n++){ M(2-.5*(n+.2),0) L(2-.5*(n+.4),0) } fprintf(o,".06 W 1 .5 0 RGB S\n");
for(n=0;n<21;n++){ M(2-.5*(n+.7),0) L(2-.5*(n+.9),0) } fprintf(o,".06 W 0 .5 1 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
       system("epstopdf Esqrt2iterMap.eps"); 
       system(    "open Esqrt2iterMap.pdf"); //for macintosh
}

Latex generator of labels

% %<br> % Copyleft 2012 by Dmitrii Kouznetsov %<br> \documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \usepackage{graphicx} %<br> \usepackage{rotating} %<br> \paperwidth 526pt %<br> \paperheight 520pt %<br> \topmargin -103pt %<br> \oddsidemargin -95pt %<br> \textwidth 1100pt %<br> \textheight 1100pt %<br> \pagestyle {empty} %<br> \newcommand \sx {\scalebox} %<br> \newcommand \rot {\begin{rotate}} %<br> \newcommand \ero {\end{rotate}} %<br> \newcommand \ing {\includegraphics} %<br> \begin{document} %<br> \sx{2.5}{ \begin{picture}(208,205) %<br> % \put(6,5){\ing{arctaniacontour}} %<br> % \put(6,5){\ing{sqrt2figf45e}} %<br> % \put(6,5){\ing{sqrt2figL45e}} %<br> \put(6,5){\ing{Esqrt2iterMap}} %<br> \put(2,203.4){\sx{.7}{$y$}} %<br> \put(2,184){\sx{.6}{$8$}} %<br> \put(2,164){\sx{.6}{$6$}} %<br> \put(2,144){\sx{.6}{$4$}} %<br> \put(2,124){\sx{.6}{$2$}} %<br> \put(2,104){\sx{.6}{$0$}} %<br> \put(-3,84){\sx{.6}{$-2$}} %<br> \put(-3,64){\sx{.6}{$-4$}} %<br> \put(-3,44){\sx{.6}{$-6$}} %<br> \put(-3,24){\sx{.6}{$-8$}} %<br> \put(-2,00){\sx{.6}{$-10$}} %<br> \put( 22,0){\sx{.6}{$-8$}} %<br> \put( 42,0){\sx{.6}{$-6$}} %<br> \put( 62,0){\sx{.6}{$-4$}} %<br> \put( 82,0){\sx{.6}{$-2$}} %<br> \put(106,0){\sx{.6}{$0$}} %<br> \put(126,0){\sx{.6}{$2$}} %<br> \put(146,0){\sx{.6}{$4$}} %<br> \put(166,0){\sx{.6}{$6$}} %<br> \put(186,0){\sx{.6}{$8$}} %<br> \put(203,0){\sx{.7}{$x$}} %<br> %<br> \put(056,103.5){\sx{.99}{\bf cut}} %<br> %<br> \put(179,144.5){\sx{.99}{\rot{-18}$v\!=\!6$\ero}} %<br> \put(175,131.5){\sx{.99}{\rot{-12}$v\!=\!4$\ero}} %<br> \put(173,118){\sx{.99}{\rot{-7}$v\!=\!2$\ero}} %<br> \put(172,103.5){\sx{.99}{$v\!=\!0$}} %<br> \put(173,088.7){\sx{.99}{\rot{6}$v\!=\!-2$\ero}} %<br> \put(176,075.7){\sx{.99}{\rot{9}$v\!=\!-4$\ero}} %<br> %<br> \put(087,128){\sx{.99}{\rot{48}$u\!=\!-1$\ero}} %<br> \put(103,120){\sx{.99}{\rot{64}$u\!=\!0$\ero}} %<br> \put(118,115){\sx{.99}{\rot{74}$u\!=\!1$\ero}} %<br> \put(130,110){\sx{.99}{\rot{79}$u\!=\!2$\ero}} %<br> \put(149.6,105){\sx{.99}{\rot{82}$u\!=\!4$\ero}} %<br> \put(166,103){\sx{.99}{\rot{84}$u\!=\!6$\ero}} %<br> \end{picture} %<br> } %<br> \end{document}