Fibonacci as iteration of fractional linear function

[Gottfried]

Unfortunately it is not a nice continuous iteration, because for non-integer t we have an imaginary part (due to \(\Psi\) being negative). This has to do with the pole in the middle between the two fixed points which would not allow for e.g. a real half iterate.

Would it be good to have a "dual"-to-f(x)-function which is real on the real argument, and where \( f_d^{o2k}(x)=f^{o2k}(x) \) but \( f_d^{o1+2k}(x) \ne f^{o1+2k}(x) \) ?  
Perhaps \( f_d(x) \) is better suited for your discussion?

See the pink line for \( f^h(x) \) and the blue line for \( f_d^h(x) \) starting at \(x=1\) for \( h=0..4 \)