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.

The following picture of the upper part of the Shell-Thron region was made by Andrew.

This picture gives a combined view of the locus of the fixpoints, their logs (being on the unit-circle) and the corresponding complex bases: (the symbol $\lambda$ is replaced by the letter $t$ )