Fibonacci as iteration of fractional linear function

[JmsNxn]

Is there a way to perform, let's say, "a crescent iteration" on the fibonacci sequence, so that we somehow map it to the reals?

We can always multiply by a 1-periodic function \(\theta(z)\), would it be possible to make \(\theta(z)F(z)\) real valued?

I've always wondered that, but I could never think of a solution; now seems as good a time to ask as any.

I thought Gottfried has the answer to that?

 Oh yes, my mistake. I didn't even see Gottfried's post. I just saw your reply to me. I'm curious what \(\theta\) would look like though. I apologize again.

But there's still a bit of work involved, here. We have a real iterate of \(1/(1+x)\), but how exactly would we pull out the Fibonacci formula?

Would we have to solve:

\[
f_d(t) = \frac{F_{\text{Gottfried}}(t) - F_{\text{Gottfried}}(t-1)z}{F_{\text{Gottfried}}(t+1) - F_{\text{Gottfried}}(t)z}\\
\]

Or can you see a simpler way that I'm not seeing, where \(F_{\text{Gottfried}}(t) = \theta(t)\phi(t)\)?