Half-iterate exp(z)-1: hypothese on growth of coefficients

[bo198214]

I'd continue to contradict baker. So please show me evidences that showing there ain't a single function that's holomorphic about \(z=0\).

You can have a look into the paper of Baker.
It has something to do with that the parabolic fixed point is a point of non-normality (Julia set) and that the Julia set is a perfect set - means it has no isolated points.
I attach the file here for archiving purpose.

The general theory there is anyways that there are these Leau-Fatou-flower petals around a fixed point of multiplier 1, where you can define Abel-Function (or also called Fatou-Coordinates), hence you have fractional iterates there. And these are *asymptotically* analytic in the fixed point, i.e. the powerseries coefficients converge to those of the formal solution at the fixed point. But the normal thing is that these Abel-Functions/Fatou-Coordinates are different from each other (and hence the iterates). Only in exceptional cases (e.g. LFTs) they agree.