Fractals from calculations of 2^I, 2^(2^I), 2^(2^(2^I).. a^(a^(...a^I)

[Ivars]

If we continiue like this, then we can check:
tetration of \( h(e^{-1})=\Omega \)

Here we have \( \ln(\Omega)=- \Omega. \) : Interestingly, the same logarithms appeared here where I changed the issue by always using module of ln(x):

Iterations of ln(mod(x))

Anyway, if we take logarithms of negative numbers via complex numbers, we again end at \( -W(-1) \):

Everything ends at that point. But also infinitely iterated logarithm of any number AT ALL if allowed to go into complex number when it first gets negative seems to end at this point(s). Is it true, or just oscillations die out so fast that all numbers give the same result... Anyway, why exactly close to -W(-1) which reminds me of Eulers Lambert function -W(-z)? His work about it is probably only in Latin.

Jaydfox reffered to these 2 fixed points here:Imaginary Iterates of exponentiation

This number \( =0.318131505... \pm 1.337235701...I) = -W(-1) \) and \( \Omega=0.567143 =W(1) \) are kind of symmetric since they are solutions of equations:

\( e^z=z \) and
\( e^{(-z)} = z \) respectively.

Also,
\( -W(-1)^{1/-W(-1)}=e \)

\( W(1)^{1/W(1)}={1/e} \)

Ivars