Fractional calculus and tetration

[JmsNxn]

My extension \( F \) is also the sole extension that is bounded by \( |F(z)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|} \) where \( \rho, \alpha, C \in \mathbb{R}^+ \) and \( \alpha < \pi/2 \).

The regular iteration for bases \( 1<b<\eta \) satisfies that, as it is periodic and bounded in the right halfplane.
What complex bases does it work for? Does it work for base eta?

Quite literally only for those real bases so far. I'm thinking there might be a way to retrieve it for other bases, but that would require a lot of generalizing on the bare machinery I have now--making the fractional calculus techniques apply on repelling fixed points. All in all, the method I have only works for \( 1<b<\eta \), and since the usual iteration is periodic and bounded like that, it must be mine as well. Currently I'm looking at how more regularly behaved functions look when they're iterated using FC--maybe that'll help me draw some more conclusions.