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\begin{document}
\sx{1.44}{\begin{picture}(340,340)
\put(20,20){\ing{AteSuFacMap}}
\put(14,320){\sx{1.1}{\(y\)}}
\put(14,220){\sx{1.1}{\(1\)}}
\put(14,120){\sx{1.1}{\(0\)}}
\put(4,020){\sx{1.1}{\(-1\)}}
\put(22,010){\sx{1.1}{\(0\)}}
\put(122,010){\sx{1.1}{\(1\)}}
\put(222,010){\sx{1.1}{\(2\)}}
\put(320,013){\sx{1.2}{\(x\)}}
\put(33,150){\sx{1.4}{\rot{-13}\(v\!=\!0.1\)\ero}} 
\put(40,120){\sx{1.4}{\rot{0}\(v\!=\!0\)\ero}} 
\put(36,91){\sx{1.4}{\rot{11}\(v\!=\!-0.1\)\ero}} 

\put(29,208){\sx{1.6}{\rot{36}\(u\!=\!1\)\ero}} 
\put(175,104){\sx{1.6}{\rot{90}\(u\!=\!2\)\ero}} 

\end{picture}}

Cmplex map of combination of natural ArcTetration and SuperFactorial:

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

The  map  is shown in the top picture with

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

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

in the \(x,y\) plane.

\vskip 8pt
\sx{1.4}{\bf References}
\vskip 4pt

\url{https://mizugadro.mydns.jp/BOOK/468.tex}
D.Kouznetwov. Superfunctions. Lambert Academic Publishing, 2020 
\vskip 4pt

\url{http://www.vmj.ru/articles/2010_2_4.pdf}
D.Kouznetsov. Tetration as special function. Vladikavkaz Mathematical Journal, 2010, v.12, issue 2, p.31-45, In Rusian.  English version:
\url{https://mizugadro.mydns.jp/PAPERS/2009vladie.pdf} 

\end{document}
