File:Expitemap.jpg

Complex maps of iterates of natural exponent;

$u\!+\!\mathrm i v=\exp^n(x\!+\!\mathrm i y)$

Usage: this is figure 15.5 of the book Суперфункции (2014, In Russian) ; the English version is in preparation in 2015.

Routines to evaluate the non–integer iterates are described also in the Vladikavkaz Matehmatical Journal .

C++ generator of the First map
As the codes to generate the maps are very similar, I load the only generator of the first map. Files ado.cin, conto.cin, fsexp.cin, fslog.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, cu,cd; int M=401,M1=M+1; int N=201,N1=N+1; DB X[M1],Y[N1]; DB *g, *f, *w; // w is working array. g=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); f=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); w=(DB *)malloc((size_t)((M1*N1)*sizeof(DB))); char v[M1*N1]; // v is working array FILE *o;o=fopen("exp10map.eps","w"); ado(o,802,402); fprintf(o,"401 1 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); DO(m,M1) X[m]=-4.+.02*(m-.5); DO(n,N1) Y[n]=0.+.02*(n-.5); // for(n=0;n.019) { //    c=exp(z); c=FSEXP(1.+FSLOG(z)); p=Re(c); q=Im(c); // if(p>-12 && p<12 && fabs(q)>1.e-12) g[m*N1+n]=p; // if(q>-12 && q<12 && fabs(q)>1.e-12) f[m*N1+n]=q; }       }} fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=2.;q=1; for(m=-8;m<8;m++)for(n=2;n<10;n+=2)conto(o,f,w,v,X,Y,M,N,(m+.1*n),-q,q);fprintf(o,".007 W 0 .6 0 RGB S\n"); for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q,q);fprintf(o,".007 W .9 0 0 RGB S\n"); for(m=0;m<8;m++) for(n=2;n<10;n+=2)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q,q);fprintf(o,".007 W 0 0 .9 RGB S\n"); for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);fprintf(o,".02 W .8 0 0 RGB S\n"); for(m= 1;m<17;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);fprintf(o,".02 W 0 0 .8 RGB S\n"); conto(o,f,w,v,X,Y,M,N, (0. ),-p,p); fprintf(o,".02 W .5 0 .5 RGB S\n"); for(m=-16;m<17;m++)conto(o,g,w,v,X,Y,M,N,(0.+m),-p,p);fprintf(o,".02 W 0 0 0 RGB S\n");
 * 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 "conto.cin"
 * 5) include "fsexp.cin"
 * 6) include "fslog.cin"

conto(o,f,w,v,X,Y,M,N, 1.3372357014306895, -p,p);fprintf(o,".005 W 0 0 0 RGB S\n"); conto(o,g,w,v,X,Y,M,N, .31813150520476413, -p,p);fprintf(o,".005 W 0 0 0 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n"); fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); free(f); free(g); free(w); system("epstopdf exp10map.eps"); system(   "open exp10map.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex combiner
\documentclass[12pt]{article} \usepackage{graphicx} \usepackage{rotating} \usepackage{geometry} \paperwidth 438px %\paperheight 134px \paperheight 682px \topmargin -107pt \oddsidemargin -84pt \textheight 800px \pagestyle{empty} \begin{document} \newcommand \ing {\includegraphics} \newcommand \sx {\scalebox}

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

\newcommand \LeLa { \put(-24,384){\sx{4}{$y$}} \put(-24,288){\sx{4}{$3$}} \put(-24,188){\sx{4}{$2$}} \put(-24,88){\sx{4}{$1$}} \put(-24,-12){\sx{4}{$0$}} } \newcommand \DoLa { \put(70,-36){\sx{3.8}{$-3$}} \put(170,-36){\sx{3.8}{$-2$}} \put(270,-36){\sx{3.8}{$-1$}} \put(390,-36){\sx{3.8}{$0$}} \put(490,-36){\sx{3.8}{$1$}} \put(590,-36){\sx{3.8}{$2$}} \put(690,-36){\sx{3.8}{$3$}} \put(786,-36){\sx{3.8}{$x$}} } %\begin{figure}%[H] ~ %\sx{.294}{\begin{picture}(802,402) \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp10map}} \LeLa \put(20,342){\sx{7}{$n\!=\!1$}} \put(46,300){\rot{0}\sx{5}{$v\!=\!0$} \ero} \put(50,144){\rot{0}\sx{5}{$u\!=\!0$} \ero} \put(46,-13){\rot{0}\sx{5}{$v\!=\!0$} \ero} \put(260,220){\rot{90}\sx{4.5}{$u\!=\!-0.2$} \ero} \put(424,224){\rot{90}\sx{5}{$u\!=\!-1$} \ero} \put(244,4){\rot{56}\sx{4.5}{$u\!=\!0.2$} \ero} \put(404,4){\rot{56}\sx{5}{$u\!=\!1$} \ero} \put(474,4){\rot{56}\sx{5}{$u\!=\!2$} \ero} \end{picture}} ~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm10map}} \put(20,340){\sx{7}{$n\!=\!-1$}}

\put(30,144){\rot{56} \sx{5}{$u\!=\!1.4$}\ero} %%%%%% \put(100,114){\rot{55} \sx{5}{$u\!=\!1.2$}\ero} %%%%%% \put(160,102){\rot{54} \sx{5}{$u\!=\!1$}\ero}

\put(2,42){\rot{-8} \sx{5}{$v\!=\!3$}\ero} \put(546,196){\rot{56} \sx{5}{$v\!=\!1$}\ero} \put(580,162){\rot{45} \sx{5}{$v\!=\!0.8$}\ero} \put(610,122){\rot{34} \sx{5}{$v\!=\!0.6$}\ero} \put(626,79){\rot{22} \sx{5}{$v\!=\!0.4$}\ero} \put(640,34){\rot{11} \sx{5}{$v\!=\!0.2$}\ero} \put(641,-12){\rot{0} \sx{5}{$v\!=\!0$}\ero} \put(8,-13){\sx{5}{\bf cut}} \end{picture}} \vskip 8pt

~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp09map}} \LeLa \put(20,340){\sx{7}{$n\!=\!0.9$}} \put(50,202){\rot{20}\sx{5}{$v\!=\!0$} \ero} \put(400,182){\rot{56}\sx{5}{$v\!=\!1$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(178,2){\rot{54}\sx{5}{$u\!=\!0$} \ero} \put(430,146){\rot{7}\sx{5}{$u\!=\!0$} \ero} \put(432, 46){\rot{43}\sx{5}{$u\!=\!1$} \ero} \put(482, 10){\rot{52}\sx{5}{$u\!=\!2$} \ero} \end{picture}} ~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm09map}} \put(20,340){\sx{7}{$n\!=\!-0.9$}} \put(180,198){\rot{27} \sx{5}{$u\!=\!1$} \ero} \put(270,14){\rot{47} \sx{5}{$u\!=\!0$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(606,250){\rot{48} \sx{5}{$v\!=\!1$} \ero} \put(636,134){\rot{29} \sx{5}{$v\!=\!0.6$} \ero} \put(644,84){\rot{19} \sx{5}{$v\!=\!0.4$} \ero} \put(648,32){\rot{10} \sx{5}{$v\!=\!0.2$} \ero} \put(649,-13){\rot{0} \sx{5}{$v\!=\!0$}\ero} \put(8,-13){\sx{5}{\bf cut}} \end{picture}} \vskip 8pt

~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp08map}} \LeLa \put(20,340){\sx{7}{$n\!=\!0.8$}} \put(168,198){\rot{35}\sx{5}{$v\!=\!0$} \ero} \put(390,182){\rot{66}\sx{5}{$v\!=\!1$} \ero} \put(430,146){\rot{13}\sx{5}{$u\!=\!0$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(40,0){\rot{46}\sx{3.8}{$u\!=\!-0.2$} \ero} \put(242,2){\rot{54}\sx{5}{$u\!=\!0$} \ero} \put(432, 42){\rot{46}\sx{5}{$u\!=\!1$} \ero} \put(488, 1){\rot{59}\sx{5}{$u\!=\!2$} \ero} \end{picture}} ~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm08map}} \put(20,340){\sx{7}{$n\!=\!-0.8$}} \put(184,192){\rot{27} \sx{5}{$u\!=\!1$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(230,36){\rot{34} \sx{5}{$u\!=\!0$} \ero} \put(6,1){\rot{52} \sx{4}{$u\!=\!0.4$} \ero} \put(166,3){\rot{54} \sx{4}{$u\!=\!0.2$} \ero} \put(506,234){\rot{53} \sx{4}{$v\!\approx\!1.337$} \ero} \put(606,220){\rot{42} \sx{5}{$v\!=\!1$} \ero} \put(636,120){\rot{26} \sx{5}{$v\!=\!0.6$} \ero} \put(644,76){\rot{16} \sx{5}{$v\!=\!0.4$} \ero} \put(648,30){\rot{8} \sx{5}{$v\!=\!0.2$} \ero} \put(8,-13){\sx{5}{\bf cut}} \end{picture}} \vskip 8pt

~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp05map}} \LeLa \put(20,340){\sx{7}{$n\!=\!0.5$}} \put(360,294){\rot{11} \sx{5}{$u\!=\!-2$} \ero} \put(382,222){\rot{22} \sx{5}{$u\!=\!-1$} \ero} \put(420,146){\rot{38} \sx{5}{$u\!=\!0$} \ero} \put(490,99){\rot{54} \sx{5}{$u\!=\!1$} \ero} \put(550,82){\rot{58} \sx{5}{$u\!=\!2$} \ero} \put(600,66){\rot{62} \sx{5}{$u\!=\!3$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(680,90){\rot{-16} \sx{5}{$v\!=\!3$} \ero} \put(680,24){\rot{-8} \sx{5}{$v\!=\!1$} \ero} \put(210,154){\rot{78} \sx{5}{$v\!=\!0$} \ero} \end{picture}} ~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm05map}} \put(20,340){\sx{7}{$n\!=\!-0.5$}} \put(602,386){\rot{-61} \sx{5}{$u\!=\!2$} \ero} \put(382,302){\rot{-41} \sx{5}{$u\!=\!1$} \ero} \put(20,272){\rot{-6} \sx{4.5}{$u\!\approx\!0.318$} \ero} \put(82,206){\rot{-3} \sx{5}{$u\!=\!0$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(486,298){\rot{42} \sx{5}{$v\!=\!2$} \ero} \put(680,190){\rot{19} \sx{5}{$v\!=\!1$} \ero} \put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero} \put(8,-13){\sx{5}{\bf cut}} \end{picture}} \vskip 8pt

~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp02map}} \put(20,340){\sx{7}{$n\!=\!0.2$}}  \LeLa %\DoLa \put(353,224){\rot{60} \sx{5}{$u\!=\!-1$} \ero} \put(452,224){\rot{66} \sx{5}{$u\!=\!0$} \ero} \put(530,200){\rot{71} \sx{5}{$u\!=\!1$} \ero} \put(602,174){\rot{73} \sx{5}{$u\!=\!2$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(680,328){\rot{-15} \sx{5}{$v\!=\!5$} \ero} \put(680,260){\rot{-11} \sx{5}{$v\!=\!4$} \ero} \put(680,194){\rot{-10} \sx{5}{$v\!=\!3$} \ero} \put(680,125){\rot{-7} \sx{5}{$v\!=\!2$} \ero} \put(680,56){\rot{-3} \sx{5}{$v\!=\!1$} \ero} \put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero} \put(200,240){\rot{-31} \sx{4.6}{$v\!\approx\!1.337$} \ero} \end{picture}} ~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm02map}} \put(20,340){\sx{7}{$n\!=\!-0.2$}}  %\DoLa \put(158,312){\rot{-57} \sx{5}{$u\!=\!-1$} \ero} \put(302,326){\rot{-66} \sx{5}{$u\!=\!0$} \ero} \put(436,346){\rot{-71} \sx{5}{$u\!=\!1$} \ero} \put(566,388){\rot{-74} \sx{5}{$u\!=\!2$} \ero} \put(10,122){\sx{5}{\bf cut}} %\put(680,348){\rot{5} \sx{5}{$v\!=\!3$} \ero} \put(680,268){\rot{9} \sx{5}{$v\!=\!2$} \ero} \put(470,132){\rot{8} \sx{4.6}{$v\!\approx\!1.337$} \ero} %\put(470,128){\rot{4.6} \sx{5}{$v\!\approx\!1.3372357$} \ero} \put(680,124){\rot{4} \sx{5}{$v\!=\!1$} \ero} \put(680,-14){\rot{0} \sx{5}{$v\!=\!0$} \ero} \put(8,-13){\sx{5}{\bf cut}} \end{picture}}\vskip 8pt

~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exp01map}} \put(20,340){\sx{7}{$n\!=\!0.1$}}  \LeLa \DoLa \put(340,228){\rot{74} \sx{5}{$u\!=\!-1$} \ero} \put(438,228){\rot{79} \sx{5}{$u\!=\!0$} \ero} \put(526,214){\rot{81} \sx{5}{$u\!=\!1$} \ero} \put(614,200){\rot{83} \sx{5}{$u\!=\!2$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(680,318){\rot{-6} \sx{5}{$v\!=\!4$} \ero} \put(680,237){\rot{-5} \sx{5}{$v\!=\!3$} \ero} \put(680,154){\rot{-4} \sx{5}{$v\!=\!2$} \ero} \put(680,72){\rot{-3} \sx{5}{$v\!=\!1$} \ero}

\put(114,210){\rot{-19} \sx{4.6}{$v\!\approx\!1.337$} \ero} \end{picture}} ~ \sx{.26}{\begin{picture}(802,402) \put(0,0){\ing{exm01map}} \put(20,340){\sx{7}{$n\!=\!-0.1$}}  \DoLa \put(228,330){\rot{-74} \sx{5}{$u\!=\!-1$} \ero} \put(346,352){\rot{-80} \sx{5}{$u\!=\!0$} \ero} \put(462,366){\rot{-82} \sx{5}{$u\!=\!1$} \ero} \put(576,386){\rot{-81} \sx{5}{$u\!=\!2$} \ero} \put(10,122){\sx{5}{\bf cut}} \put(680,348){\rot{5} \sx{5}{$v\!=\!3$} \ero} \put(680,224){\rot{4} \sx{5}{$v\!=\!2$} \ero} \put(470,128){\rot{4} \sx{4.6}{$v\!\approx\!1.337$} \ero} %\put(470,128){\rot{4.6} \sx{5}{$v\!\approx\!1.3372357$} \ero} \put(680,104){\rot{3} \sx{5}{$v\!=\!1$} \ero} \put(8,-13){\sx{5}{\bf cut}} \end{picture}}

\end{document}