Fibonacci as iteration of fractional linear function

[bo198214]

I thought it was going to be one of those \(f^{\circ 2 \circ 1/2} \neq f\) problems, but I guess I got blinded like a moth to the flame. Thinking there was the solution. lol.

You thought absolutely right. And this \(f^{\circ 2\circ 1/2}\) is the function that only coincides with every second fibonacci number.

I really believe there must be a "crescent iteration" fibonacci. It's probably not injective on \(\mathbb{R}^+\), but there's gotta be one. It just seems to simple. We're just looking for a \(\theta(z)\phi(z)\) that is real valued.

You (and Sheldon) are the ones with the numerical equipment of tinkering with the theta function, isn't it?

Pole or not but at the right fixed point the function looks very similar to the eta minor case - I mean in the sense of negative multiplier. And there you could derive a real super function, so why not here ...