Fibonacci as iteration of fractional linear function

[Leo.W]

Oh I see u meant eta minor as \(e^{-e}\) I thought it was \(e^{\frac{1}{e}}_-\)

That's eta proper (or maybe eta major) \(\eta=e^{\frac{1}{e}}\) - that's how we call it on this forum
- Yeah thx bo

Albeit these superfunctions would oscillate around the fixed point as a limit at infty, thus uninvertible, and thus would not grant you for \(f^s\circ f^t=f^{s+t}\), they're contradicts.

Well, then its not a superfunction, isn't it?! Just don't know whether James superfunction is a real superfunction ... this oscillating behaviour looked very similar to James function.

nope, the contradicts happen because things like the way u may wish a real-to-real (-1)^x for real x
For most cases even the multiplier is negative, we can still get a superfunction because it's complex-to-complex\
And meanwhile the superfunction only guarantees \(F(z+1)=T(F(z))\) for some T, not \(F(z+t)=T^t(F(z))\) for all real or even complex t. So these examples indeed are superfunctions but wont always allow you to have \(f^s\circ f^t=f^{s+t}\)
I interpret your post as to find a superfunction that is real-to-real and also preserve the property \(f^s\circ f^t=f^{s+t}\), and in that sense it's impossible though, you can take it as extending (-1)^x to a real-to-real function while preserving \((-1)^s(-1)^t=(-1)^{s+t}\)
if my interpretation donot commersurate with your original intention just ignore me lol
It's 2am now I better go sleep