File:SuVorMap150.jpg

Complex map of the SuperVoros function, SuVor
is shown with
lines \(u=\Re(\mathrm{SuVor}(z)\) and
lines \(v=\Im(\mathrm{SuVor}(z)\)
in plane \(z=x\!+\!\mathrm i y\)

SuVor is solution of the Transfer equation

\( \mathrm{SuVor}(z\!+\!1)=\mathrm{Voros}\big( \mathrm{SuVor}(z)\big) \)

C++ generator of curves

Files conto.cin and ado.cin (with lowercase letters) should be loaded in order to compile the code below.

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
#include "conto.cin"

z_type T(z_type z){ return 2.*z+z*z;}
z_type F0(z_type z){ return exp(log(2.)*z);}
z_type F1(z_type z){  z_type e=exp(log(2.)*z); return e+.5*e*e;}

int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
  int M=201,M1=M+1;
  int N=201,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("VorosMa.eps","w");ado(o,608,1008);
FILE *o;o=fopen("SuVorF1Ma.eps","w");ado(o,608,1008);
fprintf(o,"304 504 translate\n 100 100 scale\n");
DO(m,M1) X[m]=-3.+.03*(m-.5);
DO(n,N1) Y[n]=-5.+.05*(n-.5);

for(m=-3;m<4;m++){if(m==0){M(m,-3.04)L(m,3.04)} else{M(m,-5)L(m,5)}}
for(n=-5;n<6;n++){ M( -3 ,n)L(3,n)}
fprintf(o,".008 W 0 0 0 RGB 2 setlinecap S\n");

// maq(3.);
DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;}
DO(m,M1){x=X[m]; //printf("%5.2f\n",x);
DO(n,N1){y=Y[n]; z=z_type(x,y);
// c=F(.5+E(z));
//        c=-1.8*exp(sqrt(2.)*log(-z));
//        c=2.*z+z*z;
        c=F1(z);
        p=Re(c);q=Im(c);
//        if(p>-4.9 && p<4.9) 
        {g[m*N1+n]=p;}
//        if(q>-4.9 && q<4.9 && fabs(q)>1.e-11 ) 
        {f[m*N1+n]=q;}
                        }}

fprintf(o,"1 setlinejoin 2 setlinecap\n");
//p=.8;q=.4;
p=4.;q=3;
//#include"plof.cin"
for(m=-4;m<4;m++)
for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".005 W 0 .6 0 RGB S\n");
for(m=0;m<4;m++)
for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".005 W .9 0 0 RGB S\n");
for(m=0;m<4;m++)
for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".005 W 0 0 .9 RGB S\n");

for(m= 1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p); fprintf(o,".011 W .9 0 0 RGB S\n");
for(m= 1;m<9;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".011 W 0 0 .9 RGB S\n");
for(m=-8;m<9;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p); fprintf(o,".010 W 0 0 0 RGB S\n");

                  conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".011 W .6 0 .6 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
        system("epstopdf SuVorF1Ma.eps"); // both linux and mac
//      system(    "open SuVorF1Ma.pdf"); // for mac
//      system(    "xpdf SuVorF1Ma.pdf"); // for linux
}

Latex generator of labels

File VorosMa.pdf should be generated with code above in order to compile the latex document below:

\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 640pt
\paperheight 1030pt
\textwidth 700pt
\textheight 1700pt
\usepackage{graphics}
\usepackage{rotating}
\pagestyle{empty}
\parindent 0pt
\topmargin -108pt
\oddsidemargin -72pt
\newcommand \ing {\includegraphics}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\begin{document}% 
\begin{picture}(638,1028)
%\put(30,20){\ing{SuVorF1ma.pdf}}
\put(30,20){\ing{SuVorF1p5ma.pdf}}
\put(14,1015){\sx{2.4}{$y$}}
\put(14,916){\sx{2.4}{$4$}}
\put(14,816){\sx{2.4}{$3$}}
\put(14,716){\sx{2.4}{$2$}}
\put(14,616){\sx{2.4}{$1$}}
\put(14,516){\sx{2.4}{$0$}}
\put(-4,416){\sx{2.4}{$-1$}}
\put(-4,316){\sx{2.4}{$-2$}}
\put(-5,216){\sx{2.4}{$-3$}}
\put(-5,116){\sx{2.4}{$-4$}}
\put(-5, 16){\sx{2.4}{$-5$}}
\put(11,0){\sx{2.4}{$-3$}}
\put(111,0){\sx{2.4}{$-2$}}
\put(211,0){\sx{2.4}{$-1$}}
\put(330,0){\sx{2.4}{$0$}}
\put(430,0){\sx{2.4}{$1$}}
\put(530,0){\sx{2.4}{$2$}}
%\put(630,0){\sx{2.4}{$3$}}
\put(624,3){\sx{2.4}{$x$}}

\put(44,972){\sx{3}{$v\!=\!0$}}

\put(44,735){\rot{-4}\sx{3}{$u\!=\!0$}\ero}
\put(54,635){\rot{26}\sx{3}{$u\!=\!0.1$}\ero}

\put(54,612){\rot{-35}\sx{3}{$v\!=\!0.1$}\ero}
\put(44,517){\sx{3}{$v\!=\!0$}}
\put(50,420){\rot{30}\sx{3}{$v\!=\!-0.1$}\ero}

\put(44,300){\rot{4}\sx{3}{$u\!=\!0$}\ero}
\put(68,190){\rot{37}\sx{3}{$u\!=\!-0.1$}\ero}
\put(130,128){\rot{54}\sx{3}{$u\!=\!-0.2$}\ero}
\put(188,86){\rot{68}\sx{3}{$u\!=\!-0.3$}\ero}

\put(44,62){\sx{3}{$v\!=\!0$}}

\end{picture}
\end{document}

References