File:Logiha300.jpg

Complex map of the halfiterate or the Logistic operator $T(z)\!=\!s \,z\,(z\!-\!1)$ for $s\!=\!3$, upper map; $s\!=\!4$, intermadiate; $s\!=\!5$, at bottom.

$u\!+\!\mathrm i v = \mathrm{LogisticOperator}_s^{1/2}(x\!+\!\mathrm i y)$

The function is real-holomorphic; the map are symmetric; so, for the two first cases, the only half of the map is shown.

C++ generator of the upper map
// 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=302,M1=M+1; int N=51,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("logi3e4a.eps","w");ado(o,304,204); FILE *o;o=fopen("logihalf3.eps","w");ado(o,304,104); fprintf(o,"152 2 translate\n 100 100 scale\n"); DO(m,M1) X[m]= -1.5+.02*(m-.5); DO(n,N1) Y[n]=0.+.02*(n-.5);
 * 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 "efjh.cin"

for(m=-3;m<4;m++){if(m==0){M(.5*m,0)L(.5*m,1.04)} else{M(.5*m,0)L(.5*m,1)}} for(n=0;n<3;n++){         M( -1.5 ,.5*n)L(1.5,.5*n)} fprintf(o,".004 W 0 0 0 RGB S\n");

//maq(3.); 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=E(H(z))-1.; //     c=F(1.+E(0.1*z)); //     c=F(z); c=F(.5+E(z)); c=F(.5+E(c)); //     c=boe(z); //     c=.5*(1.-cos(exp((z+1.)/LQ))); //     d=H(F(z-1.)); //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.; //     if(p>-99 && p<20) g[m*N1+n]=p; p=Re(c);q=Im(c); if(p>-99 && p<99)      {g[m*N1+n]=p;} if(q>-99 && q<99)      {f[m*N1+n]=q;} //     if(q>-99 && q<99 && fabs(q)>1.e-11 )    {f[m*N1+n]=q;} }}

fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=2;q=.4; //#include"plof.cin" for(m=-2;m<2;m++) for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 .6 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".002 W .9 0 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 0 .9 RGB S\n");

for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);  fprintf(o,".006 W .9 0 0 RGB S\n"); for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".006 W 0 0 .9 RGB S\n"); for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".006 W 0 0 0 RGB S\n");

conto(o,f,w,v,X,Y,M,N, (0. ),-99,99);  fprintf(o,".008 W .5 0 .5 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n");

//M(-3.02,0)L(0,0) //M(1.-1./Q,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); M(0.7,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); //for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)} for(n=0;n<12;n++) {    M(0.7+.07*(n+.3),0) L(0.7+.07*(n+.7),0)} fprintf(o,"0.014 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logihalf3.eps"); // for linux system(   "open logihalf3.pdf"); // for mac getchar; system("killall Preview"); }

C++ generator of the second map
// 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=302,M1=M+1; int N=51,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("logi3e4a.eps","w");ado(o,304,204); FILE *o;o=fopen("logihalf4.eps","w");ado(o,304,104); fprintf(o,"152 2 translate\n 100 100 scale\n"); DO(m,M1) X[m]= -1.5+.02*(m-.5); DO(n,N1) Y[n]=0.+.02*(n-.5);
 * 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 "efjh.cin"

for(m=-3;m<4;m++){if(m==0){M(.5*m,0)L(.5*m,1.04)} else{M(.5*m,0)L(.5*m,1)}} for(n=0;n<3;n++){         M( -1.5 ,.5*n)L(1.5,.5*n)} fprintf(o,".004 W 0 0 0 RGB S\n");

//maq(3.); maq(4.); 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=E(H(z))-1.; //     c=F(1.+E(0.1*z)); //     c=F(z); c=F(.5+E(z)); c=F(.5+E(c)); //     c=boe(z); //     c=.5*(1.-cos(exp((z+1.)/LQ))); //     d=H(F(z-1.)); //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.; //     if(p>-99 && p<20) g[m*N1+n]=p; p=Re(c);q=Im(c); if(p>-99 && p<99)      {g[m*N1+n]=p;} if(q>-99 && q<99)      {f[m*N1+n]=q;} //     if(q>-99 && q<99 && fabs(q)>1.e-11 )    {f[m*N1+n]=q;} }}

fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=2;q=.4; //#include"plof.cin" for(m=-2;m<2;m++) for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 .6 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".002 W .9 0 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 0 .9 RGB S\n");

for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);  fprintf(o,".006 W .9 0 0 RGB S\n"); for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".006 W 0 0 .9 RGB S\n"); for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".006 W 0 0 0 RGB S\n");

conto(o,f,w,v,X,Y,M,N, (0. ),-99,99);  fprintf(o,".008 W .5 0 .5 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n");

//M(-3.02,0)L(0,0) //M(1.-1./Q,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); M(.96,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); //for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)} for(n=0;n<10;n++) {    M(.96+.07*(n+.3),0) L(.96+.07*(n+.7),0)} fprintf(o,"0.014 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logihalf4.eps"); // for linux system(   "open logihalf4.pdf"); // for mac getchar; system("killall Preview"); }

C++ generator of the third map
// 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=302,M1=M+1; int N=101,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("logi3e4a.eps","w");ado(o,304,204); FILE *o;o=fopen("logihal5.eps","w");ado(o,304,204); fprintf(o,"152 102 translate\n 100 100 scale\n"); DO(m,M1) X[m]= -1.5+.02*(m-.5); DO(n,N1) Y[n]=-1.+.02*(n-.5);
 * 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 "efjh.cin"

for(m=-3;m<4;m++){if(m==0){M(.5*m,-1.04)L(.5*m,1.04)} else{M(.5*m,-1)L(.5*m,1)}} for(n=-2;n<3;n++){        M( -1.5 ,.5*n)L(1.5,.5*n)} fprintf(o,".004 W 0 0 0 RGB S\n");

//maq(3.); maq(5.); 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=E(H(z))-1.; //     c=F(1.+E(0.1*z)); //     c=F(z); c=F(.5+E(z)); c=F(.5+E(c)); //     c=boe(z); //     c=.5*(1.-cos(exp((z+1.)/LQ))); //     d=H(F(z-1.)); //     p=abs(c-d)/(abs(c)+abs(d));  p=-log(p)/log(10.)-1.; //     if(p>-99 && p<20) g[m*N1+n]=p; p=Re(c);q=Im(c); if(p>-99 && p<99)      {g[m*N1+n]=p;} if(q>-99 && q<99)      {f[m*N1+n]=q;} //     if(q>-99 && q<99 && fabs(q)>1.e-11 )    {f[m*N1+n]=q;} }}

fprintf(o,"1 setlinejoin 2 setlinecap\n"); p=2;q=.4; //#include"plof.cin" for(m=-2;m<2;m++) for(n=1;n<10;n+=1)conto(o,f,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 .6 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N,-(m+.1*n),-q, q);fprintf(o,".002 W .9 0 0 RGB S\n"); for(m=0;m<2;m++) for(n=1;n<10;n+=1)conto(o,g,w,v,X,Y,M,N, (m+.1*n),-q, q);fprintf(o,".002 W 0 0 .9 RGB S\n");

for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.-m),-p,p);  fprintf(o,".006 W .9 0 0 RGB S\n"); for(m= 1;m<5;m++) conto(o,f,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".006 W 0 0 .9 RGB S\n"); for(m=-4;m<5;m++) conto(o,g,w,v,X,Y,M,N, (0.+m),-p,p);  fprintf(o,".006 W 0 0 0 RGB S\n");

conto(o,f,w,v,X,Y,M,N, (0. ),-99,99);  fprintf(o,".008 W .5 0 .5 RGB S\n");

fprintf(o,"0 setlinejoin 0 setlinecap\n");

//M(-3.02,0)L(0,0) //M(1.-1./Q,0)L(3,0) M(1.25,0)L(3,0) fprintf(o,"0.01 W 1 1 1 RGB S\n"); //for(n=0;n<32;n++) {M(-.1*n,0)L(-.1*(n+.4),0)} for(n=0;n<4;n++) {    M(1.25+.06*(n+.3),0) L(1.25+.06*(n+.7),0)} fprintf(o,"0.014 W 0 0 0 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); system("epstopdf logihal5.eps"); // for linux system(   "open logihal5.pdf"); // for mac getchar; system("killall Preview"); }

Latex generator of the labels
\documentclass[12pt,a4paper,oneside]{book} %\newcommand \EN[1]   	% make the English version \newcommand \EN[1] –     	% suppress the English version \newcommand \RU[1]   	% make the Russian version (in this document not supported) %\newcommand \RU[1] –       	% suppress the Russian version %The Japanese version is not yet supported. While \JP is used to suppress several lines at once. \newcommand \JP[1] –    	 % ореsuppress some text %\usepackage[space]{cite }% If exist. \usepackage[utf8]{inputenc} \usepackage[T2A]{fontenc} \usepackage[russian]{babel} \usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}

\usepackage{rotating} \usepackage{hyperref} \usepackage{wrapfig} \usepackage{geometry} %\renewcommand\baselinestretch{.94} %\renewcommand\baselinestretch{.94} %\renewcommand\baselinestretch{1.15} %\renewcommand\baselinestretch{1.2} %\renewcommand\baselinestretch{1.2}

\paperwidth150mm %\paperheight 257mm \paperheight 216mm %\textwidth 175mm \textwidth 150mm %\oddsidemargin 5mm \oddsidemargin -72pt %\topmargin -17mm \topmargin -103pt \textheight 257mm \pagestyle{empty} \large

\usepackage{color} \definecolor{rose}{rgb}{1,.9,1} \renewcommand\thesection{\arabic{section}} \renewcommand\figurename{Рис.} %\renewcommand\refname{References} %\renewcommand\figurename{Fig.} \newcommand \be {\begin{eqnarray}} \newcommand \ee {\end{eqnarray} } \newcommand \sx {\scalebox} \newcommand \rme 	 %%makes the base of natural logarithms Roman font %\newcommand \rme 	%%makes the base of natural logarithms Italics font; choose one of these \newcommand \rmi 	 %%imaginary unity is always roman font \newcommand \ds {\displaystyle} \newcommand \bN {\mathbb{N}} \newcommand \bC {\mathbb{C}} \newcommand \bR {\mathbb{R}} \newcommand \cO {\mathcal{O}} \newcommand \cF {\mathcal{F}} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \nS {\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!} \newcommand \pS {{~}~{~}} \newcommand \fac {\mathrm{Factorial}} \newcommand {\rf}[1] {(\ref{#1})} \newcommand{\iL}[1] {~\label{#1}\pS \rm[#1]\nS}	%make the labels visible %\newcommand{\iL}[1] {\label{#1}}			%make the labels invisible; choose one of these options \newcommand \eL[1] {\iL{#1}\ee} \newcommand \ing \includegraphics \newcommand \tet {\mathrm{tet}} \parindent 0pt \begin{document} \JP{ \color{red} \begin{picture}(425,729.88) \put(0,0){\rule{1pt}{729.88pt}} \put(425,0){\rule{1pt}{729.88pt}} \put(0,0){\rule{426pt}{1pt}} \put(0,728.88){\rule{426pt}{1pt}} \end{picture} %\vskip -740pt \vskip -729pt \color{black} }\hskip 2pt \sx{1.32}{\begin{picture}(320,120) \put(16,16){\ing{logihalf3}} \put(8,112){\sx{1.2}{$y$}} \put(8, 64.4){\sx{1.17}{$\displaystyle \frac{1}{2}$}} \put(8, 14){\sx{1.2}{$0$}} \put(60,5){\sx{1.2}{$-1$}} \put(166,5){\sx{1.2}{$0$}} \put(266,5){\sx{1.2}{$1$}} \put(312,5){\sx{1.24}{$x$}} \put(26,49.2){\sx{1.2}{\rot{12} $v\!=\!4$ \ero}} \put(29,32){\sx{1.2}{\rot{6} $v\!=\!2$ \ero}} \put(30,15){\sx{1.2}{$v\!=\!0$}} \put(222.4,32){\sx{1.2}{\rot{90}$v\!=\!0$\ero}} \put(238,24){\sx{1.2}{\rot{58}\bf cut \ero}} \put(288,15){\sx{1.2}{\rot{0}\bf cut \ero}} \put(170,113){\sx{1.2}{\rot{-20}$u\!=\!3$\ero}} \put(161,100){\sx{1.2}{\rot{-30}$u\!=\!2$\ero}} \put(151,87){\sx{1.2}{\rot{-41}$u\!=\!1$\ero}} \put(136.6,76){\sx{1.2}{\rot{-54}$u\!=\!0$\ero}} \put(125,68){\sx{1.2}{\rot{-72}$u\!=\!-1$\ero}} \put(112,64){\sx{1.2}{\rot{-79}$u\!=\!-2$\ero}} \put(97,59){\sx{1.2}{\rot{-81}$u\!=\!-3$\ero}} \put(84,56){\sx{1.2}{\rot{-83}$u\!=\!-4$\ero}} \end{picture}} \vskip 4pt

\hskip 2pt \sx{1.32}{\begin{picture}(320,120) \put(16,16){\ing{logihalf4}} \put(8,112){\sx{1.2}{$y$}} \put(8, 64.4){\sx{1.17}{$\displaystyle \frac{1}{2}$}} \put(8, 14){\sx{1.2}{$0$}} \put(60,5){\sx{1.2}{$-1$}} \put(166,5){\sx{1.2}{$0$}} \put(266,5){\sx{1.2}{$1$}} \put(312,5){\sx{1.24}{$x$}} \put(26,40){\sx{1.2}{\rot{9} $v\!=\!4$ \ero}} \put(29,27.7){\sx{1.2}{\rot{5} $v\!=\!2$ \ero}} \put(30,15){\sx{1.2}{$v\!=\!0$}} \put(222.4,32){\sx{1.2}{\rot{90}$v\!=\!0$\ero}} \put(251,24){\sx{1.2}{\rot{69}\bf cut \ero}} \put(288,15){\sx{1.2}{\rot{0}\bf cut \ero}} \put(170,114){\sx{1.2}{\rot{-19}$u\!=\!4$\ero}} \put(162,105){\sx{1.2}{\rot{-30}$u\!=\!3$\ero}} \put(155,95){\sx{1.2}{\rot{-37}$u\!=\!2$\ero}} \put(146,85){\sx{1.2}{\rot{-43}$u\!=\!1$\ero}} \put(137,77){\sx{1.2}{\rot{-54}$u\!=\!0$\ero}} \put(127.4,72){\sx{1.2}{\rot{-69}$u\!=\!-1$\ero}} \put(117.6,65){\sx{1.2}{\rot{-75}$u\!=\!-2$\ero}}% \put(106.6,62){\sx{1.2}{\rot{-78}$u\!=\!-3$\ero}}% \put(97.6,59){\sx{1.2}{\rot{-82}$u\!=\!-4$\ero}} \end{picture}} \vskip 4pt

\hskip 2pt \sx{1.32}{\begin{picture}(320,220) \put(16,16){\ing{logihal5}} \put(8,212){\sx{1.2}{$y$}} \put(8,164.5){\sx{1.17}{$\displaystyle \frac{1}{2}$}} \put(8,114){\sx{1.2}{$0$}} \put(-2,64.5){\sx{1.16}{$\displaystyle-\frac{1}{2}$}} \put(-1,14){\sx{1.2}{$-1$}} \put(60,5){\sx{1.2}{$-1$}} \put(166,5){\sx{1.2}{$0$}} \put(266,5){\sx{1.2}{$1$}} \put(312,5){\sx{1.24}{$x$}} \put(26,137){\sx{1.2}{\rot{9} $v\!=\!4$ \ero}} \put(29,125){\sx{1.2}{\rot{4} $v\!=\!2$ \ero}} \put(30,114.5){\sx{1.2}{$v\!=\!0$}} \put(28,105){\sx{1.2}{\rot{-4} $v\!=\!-2$ \ero}} \put(25,94){\sx{1.2}{\rot{-9} $v\!=\!-4$ \ero}} % \put(172,201.8){\sx{1.2}{\rot{-22} $u\!=\!4$ \ero}} \put(167,192.8){\sx{1.2}{\rot{-29} $u\!=\!3$ \ero}} \put(159.8,183.8){\sx{1.2}{\rot{-36} $u\!=\!2$ \ero}} \put(152,176){\sx{1.2}{\rot{-47} $u\!=\!1$ \ero}} % \put(82,21){\sx{1.2}{\rot{60} $u\!=\!-4$ \ero}} \put(95,18){\sx{1.2}{\rot{56} $u\!=\!-2$ \ero}} \put(110,17){\sx{1.2}{\rot{48} $u\!=\!0$ \ero}} \put(130,17){\sx{1.2}{\rot{40} $u\!=\!2$ \ero}} \put(156,17){\sx{1.2}{\rot{30} $u\!=\!4$ \ero}} % \put(263,124){\sx{1.2}{\rot{74}\bf cut \ero}} \put(298,115){\sx{1.2}{\rot{0}\bf cut \ero}} \put(222.4,102){\sx{1.2}{\rot{90}$v\!=\!0$\ero}} \end{picture}}

\end{document}