Shell-Thron region

The Shell-Thron region is the region of complex numbers $b$ where the limit of the infinite exponential tower $$\lambda=\lim_{n\to\infty} \exp_b^{\circ n}(1)$$ exists; to be explicit we mean the main branch of the power: $\exp_b(z):=b^z:=\exp(\log(b)z)$.

If this limit $\lambda$ exists then must $b^\lambda=\lambda$, i.e. $\lambda$ must be a fixpoint of $f(z)=b^z$ and $b=\lambda^{1/\lambda}$, or $$\lambda=\frac{W(-\log(b))}{-\log(b)}$$

By a result of Barrow the limit $\lambda$ exists exactly for all $b$ where $|\log(\lambda)|\le 1$.

The boundary of the Thron-Shell region, i.e. all bases $b$ where $|\log(\lambda)|=1$, is given by all $b$ where $|\exp_b'(\lambda)|=1$ due to the following rearrangements: $$\begin{align*} 1=|\exp_b'(\lambda)|=|\log(b)\exp_b(\lambda)|=|\log(b)\lambda|=|\log(\lambda^{1/\lambda})\lambda|=|\log(\lambda)| \end{align*}$$

These rearrangements are not proper derivations, but hints only, as with complex bases one always has to worry about landing in the right branch.