Uniqueness of fractionally iterated functions

[Daniel]

I understand that, Daniel. I apologize if my response seemed hostile. Wasn't my intention, lol.

All I'm saying is that the Taylor series approach, is inherently Schroder's construction. I'm not doubting that it works in any way shape or form. But I suggest you  observe your iteration \(f^{\circ 1/2}(z)\) of \(f = \sqrt{2}^z\), and trace \(z\) from \(2 \to 4\). Somewhere along that path there is a singularity (probably a tiny discontinuity/jump at about 1E-10 height). If not, you've done something wrong.

Hey, it's all good JmsNxn. I'm always happy to get thoughtful feedback, even if it is not what I have hoped for.

My construction not only encompasses the cases of Abel and Schroeder's functional equations. See my page on generating flows based on the Abel's case. I'd run your test but unfortunately I'm now without Mathematica for the first time in thirty years, so I guess I need to get good at GP-Pari.