Two Attracting Fixed Points

[tommy1729]

Is eta the only number a, such that a to the power of x has a parabolic fixed point? If not, what else could a be?

The boundary of shell thron always has a parabolic fixed point.

Let \(a \in \mathcal{S} = \{a \in \mathbb{C}\mid |\log(a)| = 1\}\). The function:

\[
f(z) = e^{\log(a)z/a}\\
\]

Which is an exponential; always has a parabolic fixed point at \(a\).

Essentially to answer your question, the boundary of the Shell-thron region is the only place you get parabolic fixed points. The value \(\eta\) is just one particular value.

yes that is correct.

I sometimes wonder about the analogue shell-thron for the double exponential functions.

In particular because there are fixpoints outside the shell-thron such as z^z = z.
( we studies that number , I posted a thread about it with title simply the complex value of the smallest nonreal z )

And also because there are half-iterates of exp(exp(x)) that are not exp(x).

I could say more , as could most regular posters here. We all have considered such ideas.

So I will leave it at that for now.

regards

tommy1729