Infinite tetration convergence

Question

I came across infinite tetrations on wikipedia (https://en.wikipedia.org/wiki/Tetration) it says that the infinite tetration converges if and only if $\ e^{-e} \leq x \leq e^{1/e}$. I was wondering if there is a proof of this.

Answer 1

This relates to the deeper questions of where does the iterates of $a^z$ converge in the complex plane for infinite exponential towers at a fixed point $A$ such that $a^A=A$ and where is the onset for period $n$ behavior.

Let $\Delta z$ be an infinitesimal. Since $a^A=A$, $a^{A+\Delta z}=A a^{\Delta z}=A + Ln A \ \! {\Delta z}$. Therefore $A+\Delta z \Rightarrow A + Ln A \ \! {\Delta z}$. The dynamics of $a^z$ in the neighborhood of a fixed point $A$ are solely dependant on the location of $A$.

Let $Ln A=e^{2 \pi i}$, then $A =e^{e^{2 \pi i}}$. But $a=A^{1/A}$, so $ a=\LARGE e^{e^{2 z i \pi-e^{2 z i \pi}}}$. Let $z=1$, then so $1.444\approx e^{1/e} = \LARGE e^{e^{2 i \pi-e^{2 i \pi}}}$. Let $z=1/2$, then so $0.065988 \approx e^{-e} = \LARGE e^{e^{i \pi-e^{i \pi}}}$

The complex function $\LARGE e^{e^{2 z i \pi-e^{2 z i \pi}}}$

The exponential Mandelbrot set by period. Red is the area of convergence or period 1, orange is period 2 and includes 0, yellow is period 3.