Fibonacci as iteration of fractional linear function

[bo198214]

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?

From the standpoint of regular iteration there is only the one (on fractional linear functions), and messing with it is blasphemy