[MSE] f(f(z)) = z , f(exp(z)) = exp(f(z))

[JmsNxn]

Tetrational Geometry

The equation \[f(exp(z)) = exp(f(z))\] tells us we want to be looking at tetration. Now add the constraint \[f(f(z))=z\] in the context of tetration. This means we are looking at period 2 tetration. The orange disk in the following image and the red disk in the image after is the area where tetration displays period 2 behavior. See Pseudocircle The number 0 lies in the disk and is period 2 under tetration,
\[0^0=1, 0^1=0\]





\[f(exp(z)) = exp(f(z)) \implies f(z) = \, ^{z+n}e\] which is inconsistent with the complex base being within the pseudocircle.
The question could be generalized to \[f(a^z) = a^{f(z)} \textrm{ and } f(f(z))=z \textrm{ where } a \textrm{ is in the pseudocircle.}\]. Since \[a \ne e\] the question has no solution beyond the trivial solution.

Fucking beautiful, Daniel. That was my suspicion; your solution is the only solution. I was trying to fiddle with shit; but I was pretty sure it was still \(z\).

Nice!