Fibonacci as iteration of fractional linear function

[Leo.W]

So what is \(\alpha^{-1}\{f\}(z)\) what is this braces notation?
What is the Abelian property?

Forgive me. Abelian means commutive, as s and t do in \(f^s\circ f^t=f^{s+t}\)

If it's continuous then it has to be a constant... so no solution if allowing the superfunction to be invertible

Yeah that was the thing with JmsNxn's superfunction at eta minor: It was not invertible, rather close to periodic.

James did many great works, yeah
but about this problem Idk if anyone can build up such superfunctions?
The eta minor case, a little similar to the kneser's tetration, they won't map \(\mathbb{R}\) to \(\mathbb{R}\) but a sub-domain of real axis. And the fixed point's multiplier were positive, not negative as ours.
It's also, not compatible and comparable to our \(f(z)=\frac{1}{1+z}\) (should take infinity into account though)
you can firstly try to build a \(f^{\frac{1}{2}}\) as an attempt, this would fail by previous analysis. It's equivalent to build a square root of -2.618.  
(lmao this expression so funny)