Fibonacci as iteration of fractional linear function

[bo198214]

And the fixed point's multiplier were positive, not negative as ours.

It is negative, that's what I mean with similar to our case.
\(f(x)=b^x\), \(b=\eta_-=e^{-e}\), parabolic fixed point \(z_0=\frac{1}{e}\), \(f'(x)=\log(b)b^x\), \(f'(z_0)=\log(b)z_0=-e\frac{1}{e}=-1\).