File:AteSuExq2mapU.png

Complex map of combination of two functions: natural ArcTetration «ate» and growing superexponential to base \(\sqrt{2}\).

\(f(z)=\mathrm{ate}\Big(\mathrm{SuExq2}(z)\Big)\)

The map is shown with

lines \(u=\Re \big(f(x\!+\!\mathrm i y)\big)\) and

lines \(v=\Im \big(f(x\!+\!\mathrm i y)\big)\)

in the \(x,y\) plane.

C++ generator of curves

/* files ado.cin, Conrec6.cin, SuExq2.cin, fslog.cin should be also loaded*/

// c++ -std=c++11 AteSuExq2map.cc -O2 -o AteSuExq2map
#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()
#define I z_type(0.,1.)
 #include"ado.cin"
 #define M(x,y) fprintf(o,"%6.4f %6.4f M\n",1.*(x),1.*(y));
 #define L(x,y) fprintf(o,"%6.4f %6.4f L\n",1.*(x),1.*(y));
#include "Conrec6.cin"
//#include "fac.cin"
//#include "SuFac.cin"
#include "SuExq2.cin" // FE

#include "fslog.cin"
//#include "filog.cin"
int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;
//int M=971,M1=M+1;
int M=601,M1=M+1;
int N=801,N1=N+1;
DB X[M1],Y[N1], g[M1*N1],f[M1*N1];
FILE *o;o=fopen("AteSuExq2map.eps","w");ado(o,612,408);
fprintf(o,"4 104 translate\n 100 100 scale\n");

DO(m,M1) X[m]=0+.01*(m-.5);
DO(n,N1) Y[n]=-1+.005*(n-.5);

for(m=0;m<7;m++){M(m,-1)L(m,3)}
for(n=-1;n<4;n++){M(0,n)L(6,n)}
fprintf(o,"2 setlinecap .008 W 0 0 0 RGB S\n");
//fprintf(o,".0007 W 0 0 0 RGB S\n");
DO(m,M1)DO(n,N1){g[m+M1*n]=9999; f[m+M1*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=Filog(z);
// c=z*z*sin(1./z);
//c=superfac(z); //SuFac
c=F45E(z); 
if(abs(c)<1.e12)
	{	c=FSLOG(c); 	}// ate
else 	{	continue;	}
p=Re(c);q=Im(c); 
if(p>-201. && p<201. && q>-201. && q<201. ){ g[m+M1*n]=p;f[m+M1*n]=q;}
       }}

printf("Try to plot\n"); // getchar();

fprintf(o,"1 setlinejoin 1 setlinecap\n"); p=.6;q=.1;

for(m=-6;m<7;m++)for(n=1;n<10;n+=1)Conrec6(o,f,X,Y,M1,N1,(m+.1*n), q); fprintf(o,".01 W 0 .6 0 RGB S\n");
for(m=0;m<7;m++) for(n=1;n<10;n+=1)Conrec6(o,g,X,Y,M1,N1,-(m+.1*n), q); fprintf(o,".01 W .9 0 0 RGB S\n");
for(m=0;m<7;m++) for(n=1;n<10;n+=1)Conrec6(o,g,X,Y,M1,N1, (m+.1*n), q); fprintf(o,".01 W 0 0 .9 RGB S\n");
for(m=1;m<11;m++) Conrec6(o,f,X,Y,M1,N1, (0.-m),p); fprintf(o,".02 W .9 0 0 RGB S\n");
for(m=1;m<11;m++) Conrec6(o,f,X,Y,M1,N1, (0.+m),p); fprintf(o,".02 W 0 0 .9 RGB S\n");
                  Conrec6(o,f,X,Y,M1,N1, (0. ),p); fprintf(o,".026 W .8 0 .8 RGB S\n");
for(m=-10;m<21;m++) Conrec6(o,g,X,Y,M1,N1, (0.+m),p); fprintf(o,".02 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
     system("epstopdf AteSuExq2map.eps"); 
     system(    "open AteSuExq2map.pdf"); //for mac
}

Latex generator of labels

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[russian]{babel}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{hyperref} 
\usepackage{rotating} 
\usepackage[export]{adjustbox}
\paperwidth 2520pt
%\paperheight 856pt
\paperheight 1712pt
%\topmargin -92pt
%\textheight 750pt
%\oddsidemargin -66pt
%\textwidth 512pt
\topmargin -148pt
\textheight 2650pt
\oddsidemargin -72pt
\textwidth 1812pt
\newcommand{\ing}{\includegraphics}
\newcommand{\sx}{\scalebox}
\newcommand{\rot}{\begin{rotate}}
\newcommand{\ero}{\end{rotate}}
\parindent 0px
\begin{document}
%\sx{.94}
\sx{4}
{\begin{picture}(640,440)
%\put(20,20){\ing{AteSuFacMap}}
\put(20,20){\ing{AteSuExq2map}}
%\put(50,370){\sx{2.7}{\(u\!+\!\mathrm i v= \mathrm{ate}\Big(\mathrm{SuExq2}(x\!+\!\mathrm i y)\Big)\)}}
\put(8,414){\sx{2}{\(y\)}}
\put(10,320){\sx{1.7}{\(2\)}}
\put(10,220){\sx{1.7}{\(1\)}}
\put(10,120){\sx{1.7}{\(0\)}}
\put(0,020){\sx{1.6}{\(-1\)}}
\put(22,4){\sx{1.7}{\(0\)}}
\put(120,4){\sx{1.7}{\(1\)}}
\put(220,4){\sx{1.7}{\(2\)}}
\put(320,4){\sx{1.7}{\(3\)}}
\put(420,4){\sx{1.7}{\(4\)}}
\put(520,4){\sx{1.7}{\(5\)}}
\put(611,6){\sx{2}{\(x\)}}
\put(40,290){\sx{2}{\rot{-40}\(v\!=\!0.1\)\ero}} 
\put(72,118.2){\sx{2}{\rot{0}\(v\!=\!0\)\ero}} 

\put(174,96){\sx{2}{\rot{90}\(u\!=\!1.6\)\ero}} 
\put(360,100){\sx{2}{\rot{90}\(u\!=\!2\)\ero}} 
\put(474,71){\sx{2}{\rot{50}\bf cut\ero}} 

\end{picture}}
\end{document}

Conjectures

In vicinity of the real axis, the levels for almost equidistant grid, as abscissa increases, the derivative approaches unity.

This observation can be formulated as a set of conjectures:

Conjecture 0

There exist real constant \(x_0\) and domain \(D\subset\mathbb C\) that includes the poisitive real axis such that \[ \mathrm {SuExq2}(z) \underset{\mathrm{ate},\ z\to \infty, \ z\in D}{\sim} \mathrm{tet}\big( z\!-\!x_0+\mathcal O(1/z)\big) \]

Conjecture 1 There exist real constant \(x_0\) and positive decreasing function \(\varepsilon\) such that \[ \lim_{x\to \infty, |y|<\varepsilon(x)} \Big( \mathrm{ate}\big(\mathrm{SuExq2}(x\!+\!\mathrm i y)\big) - \big((x\!+\!\mathrm i y) - x_0 \big)\Big) = 0 \]

Conjecture 2

For \(x>3\), \[ \mathrm{tet}(x-2) < \mathrm{SuExq2}(x) < \mathrm{tet}(x-1) \]

Conjecture 3

The constant \(x_0 \approx 1.93 \)

References