File:E1eghalfm3.jpg

Complex map of upper half iterate of exponential to base $\eta=\exp(1/\mathrm e)$ :

$u\!+\!\mathrm i v=\exp_{\eta,u}^{1/2}(x\!+\!\mathrm i y)$

where $\exp_{\eta,\mathrm u}^{1/2}(z)=$SuExp$_{\eta}\big(1/2+$SuExp$_{\eta}(z)\big)$

C++ generator of map
// files ado.cin, conto.cin, e1egf.cin, e1egi.cin should be loaded in order to compile the code below

// //using namespace std; typedef std::complex z_type; int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; int M=401,M1=M+1; int N=301,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
 * 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) define I z_type(0.,1.)
 * 4) include "e1egf.cin"
 * 5) include "e1egi.cin"
 * 6) include "conto.cin"

FILE *o;o=fopen("e1eghalf.eps","w");ado(o,402,302); fprintf(o,"101 151 translate\n 10 10 scale\n"); DO(m,M1) X[m]=-10.+.1*(m-.5); DO(n,N1) Y[n]=-15.+.1*(n-.5); for(m=-10;m<31;m++){ //if(m==0){       M(m,-10.2)L(m,10.2)} else {      M(m,-15)L(m,15)         }} for(n=-15;n<16;n++){ M(-10,n)L(30,n)} fprintf(o,".006 W 0 0 0 RGB S\n"); DO(m,M1)DO(n,N1){g[m*N1+n]=9999; f[m*N1+n]=9999;} DO(m,M1){x=X[m]; printf("run at x=%6.3f\n",x); DO(n,N1){y=Y[n]; z=z_type(x,y); c=E1EGF(.5+E1EGI(z)); //     c=E1EGF(.5+E1EGI(c)); //     d=z; //     p=abs(c-d)/abs(c+d); p=-log(p)/log(10.); p=Re(c); q=Im(c); //     if(p>-85 && p<85)  g[m*N1+n]=p; if(p>-101.3 && p<101.3 && fabs(p)> 1.e-11 && //fabs(p-1.)>1.e-9 &&         q >-101.3 && q<101.3 && fabs(q)> 1.e-11) { g[m*N1+n]=p; f[m*N1+n]=q; } }}

fprintf(o,"1 setlinejoin 1 setlinecap\n"); //p=1.;q=.5; //#include"plofu.cin" p=4.;q=.5;

for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n"); for(m= 1;m<11;m++) conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N,(0. ),-p,p);fprintf(o,".04 W .5 0 .5 RGB S\n"); for(m=-10;m<11;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");

m=15; conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n");

for(m=20; m<101;m+=10){ conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n"); } /* m=30; conto(o,f,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W .8 0 0 RGB S\n"); conto(o,f,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 .8 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(0.-m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".04 W 0 0 0 RGB S\n"); DO(m,M1){ DO(n,N1){ if( fabs(g[m*N1+n])>5. || fabs(f[m*N1+n])>5. ) { g[m*N1+n]=999.;f[m*N1+n]=999.;} }}

for(m=-5;m<6;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<5;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<5;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"); /* fprintf(o,"0 setlinecap\n"); M(-2,0)L(-8,0) fprintf(o,".08 W 1 1 1 RGB S\n"); DO(m,16){M(-2-.4*(m),0)L(-2-.4*(m+.5),0)} fprintf(o,".09 W 0 0 0 RGB S\n"); fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf e1eghalf.eps"); system(   "open e1eghalf.pdf"); //mac //     system(    "xpdf e1eghalf.pdf"); // linux getchar; system("killall Preview");// mac }

Latex generator of labels
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{rotating} \usepackage{geometry} \paperwidth 420px \paperheight 322px \topmargin -107pt \oddsidemargin -80pt \textheight 600px \pagestyle{empty} \begin{document} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox}

\newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}}

\newcommand \figaxeh { \put( -5,306){\sx{1}{$y$}} \put( -10,289){\sx{1}{$14$}} \put( -10,269){\sx{1}{$12$}} \put( -10,249){\sx{1}{$10$}} \put( -6,229){\sx{1}{$8$}} \put( -6,209){\sx{1}{$6$}} \put( -6,189){\sx{1}{$4$}} \put( -6,169){\sx{1}{$2$}} \put( -6, 149){\sx{1}{$0$}} \put(-14,129){\sx{1}{$-2$}} \put(-14,109){\sx{1}{$-4$}} \put(-14, 89){\sx{1}{$-6$}} \put(-14, 69){\sx{1}{$-8$}} \put(-18, 49){\sx{1}{$-10$}} \put(-18, 29){\sx{1}{$-12$}} \put(-18,  9){\sx{1}{$-14$}} %\put(-13, 79){\sx{1}{$-6$}} \put(400,-6.5){\sx{1}{$x$}} \put(379,-7){\sx{.9}{$28$}} \put(359,-7){\sx{.9}{$26$}} \put(339,-7){\sx{.9}{$24$}} \put(319,-7){\sx{.9}{$22$}} \put(299,-7){\sx{.9}{$20$}} \put(279,-7){\sx{.9}{$18$}} \put(259,-7){\sx{.9}{$16$}} \put(239,-7){\sx{.9}{$14$}} \put(219,-7){\sx{.9}{$12$}} \put(199,-7){\sx{.9}{$10$}} \put(181,-7){\sx{.9}{$8$}} \put(161,-7){\sx{.9}{$6$}} \put(141,-7){\sx{.9}{$4$}} \put(121,-7){\sx{.9}{$2$}} \put(101,-7){\sx{.9}{$0$}} \put( 75,-7){\sx{.9}{$-2$}} \put( 55,-7){\sx{.9}{$-4$}} \put( 35,-7){\sx{.9}{$-6$}} \put( 15,-7){\sx{.9}{$-8$}} } \begin{picture}(262,312) \put(2,2){\ing{e1eghalf}} \figaxeh %\put( 21,182){\sx{1.3}{\rot{53} \textcolor{white}{\rule{26pt}{9pt}}\ero}} \put( 18,182){\sx{1.3}{\rot{49} $v\!=\!-1$\ero}} %\put( 61,173){\sx{1.3}{\rot{85} \textcolor{white}{\rule{23pt}{9pt}}\ero}} \put( 61,175){\sx{1.3}{\rot{82} $v\!=\!0$\ero}} \put(16,150.2){\sx{1.4}{\bf cut - - - - - - -}} %\put(257,148.4){\sx{1.3}{$q\!=\!0$}} %\put(307,148.4){\sx{1.7}{$\exp_{b,3}^{[0.5]}(z)\!\rightarrow\!\infty$}} \put(316,297){\sx{1.1}{\rot{16}$u\!=\!10$\ero}} \put(323,288){\sx{1.1}{\rot{20}$u\!=\!20$\ero}} %	\put(338,274){\sx{1.1}{\rot{40}$u\!=\!40$\ero}} \put(343,266){\sx{1.1}{\rot{45}$u\!=\!50$\ero}} \put(358,254){\sx{1.1}{\rot{51}$u\!=\!70$\ero}} \put(379,242){\sx{1.1}{\rot{54}$u\!=\!100$\ero}}

\put(373,236){\sx{1.1}{\rot{-18}$v\!=\!100$\ero}} \put(357,198){\sx{1.1}{\rot{-9}$v\!=\!50$\ero}} \put(350,170){\sx{1.1}{\rot{-2}$v\!=\!20$\ero}} \put(349,160){\sx{1.1}{\rot{-1}$v\!=\!10$\ero}} \put(349,150){\sx{1.1}{$v\!=\!0$}} \put(349,141){\sx{1.1}{\rot{1}$v\!=\!-10$\ero}} \put(350,131){\sx{1.1}{\rot{2}$v\!=\!-20$\ero}} \put(357,104){\sx{1.1}{\rot{9}$v\!=\!-50$\ero}} %\put(142,225){\sx{.9}{\rot{-52} \textcolor{white}{\rule{22pt}{9pt}} \ero }} \put(166,249){\sx{1.}{\rot{-60} $v\!=\!15$\ero}} \put(144,229){\sx{1.}{\rot{-54} $v\!=\!9$\ero}} \put(130,218){\sx{1.}{\rot{-53} $v\!=\!6$\ero}} %\put( 99,175){\sx{1.2}{ \rot{45} \textcolor{white}{\rule{19pt}{10pt}} \ero }} \put( 99,174){\sx{1.2}{\rot{45} $u\!=\!0$\ero}} %\put(117,165){\sx{1.2}{\rot{57} \textcolor{white}{\rule{20pt}{9pt}} \ero }} \put(113,166){\sx{1.2}{\rot{57} $u\!=\!1$\ero}} %\put(136,155){\sx{1.2}{\rot{72} \textcolor{white}{\rule{21pt}{9pt}} \ero }} \put(136,156){\sx{1.2}{\rot{72} $u\!=\!3$\ero}} \end{picture} \end{document}