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.

See Fibonacci almost to the bottom of the page for a real iteration of the Fibonacci series.

Could you elaborate, Daniel? Sorry, I'm not too sure what's going on here.

I understand you are writing:

\[
f(z) = \sum_{n=1}^\infty f_n \frac{z^n}{n!}\\
\]

Where now we are taking a parabolic iteration:

\[
f^{\circ t}(z)\\
\]

about \(0\), but how does this produce a fractional fibonacci that is real valued?

Not doubting you, just curious.