Constructing a real valued Fibonacci iteration--its relation to \(1/1+z\)

[bo198214]

\[
\phi(t) = \frac{\theta^+(t) \Phi^t + \theta^-(t) \Psi^t}{\Phi - \Psi}\\
\]

And now we ask, where this thing is real valued.

No, we looking for a real-valued superfunction of f (as you did with eta minor).
Because the derivatives \(\frac{-1}{(1+x)^2}\) at the fixed points \(-\Phi,-\Psi\) are negative,
(also remember from here that the fixed point derivatives are the quotient of the eigenvalues, i.e. \(\frac{\Phi}{\Psi}\) and reciprocal, and \(\Psi\) is negative.)
but at least asymptotically (if not regular) you need to have \(\left(f^{\circ t}\right)'(z_0) = f'(z_0)^t \),
which means in a small vicinity it can not be real valued for non-integer t.

That's why I was so surprised about your real valued superfunction at eta minor!