Fibonacci as iteration of fractional linear function

[Gottfried]

Just to see what we are talking about with this fib2 variant:
\[\text{fib}_2(t) = \frac{\Phi^t - \left|\Psi\right|^t}{\Phi-\Psi}\]

The black dots are the real Fibonacci numbers.

Maybe it is also an option to somehow theta-distort the fib2 to fit fib. But then again, if we have one solution then there are too many more.

Just to add the reference to an interesting discussion of the Binet/complex and the real/cos()-concepts   
in MSE: https://math.stackexchange.com/questions...or-complex      
Sheldon answers on my question about this, and relates the Schroeder and the Kneser-attempt to the two methods. He gave two answers, perhaps the second answer is the more powerful one...

An answer to a later, related question gives also a plot for the real-to-real function at https://math.stackexchange.com/a/798221

Gottfried