On extension to "other" iteration roots

[bo198214]

But, Leo.W, I really have to say, you set me on a totally wrong track here, with statements like

Well bo, if you take abelian property, it'll fail then 

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.

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,

Well, in the end it lead me to find a way to handle the multi-valued case, so one could say you did something good