Fibonacci as iteration of fractional linear function

[bo198214]

However I wonder why it seems to be regular at the right fixed point ...

It's because it is not an iteration at all. It satisfies \(f^{\circ t+1} = \frac{1}{1+f^{\circ t}} \) (which corresponds to the Fibonacci identity) but it does not satisfy \(f^{\circ s+t}=f^{\circ s}\circ f^{\circ t}\), while the original Fibonacci extension is a true (and regular) iteration.
So the question still stands to find a real valued super function \(\frac{1}{1+z}\).