As announced here the motion picture of the LFT corresponding to the alternative Fibonacci extension:
The values at 0 are \(\frac{\phi_t}{\phi_{t+1}}\) and going \(\to \frac{1}{\Phi}\approx\) 0.618 for \(t\to \infty\).
One can very well see that it is not a regular iteration at the left fixed point \(z_1\), because there is no interval of length \(>1\) such that \(f^{\circ t}\) is analytic at \(z_1\) for all t in this interval.
However I wonder why it seems to be regular at the right fixed point ...
The values at 0 are \(\frac{\phi_t}{\phi_{t+1}}\) and going \(\to \frac{1}{\Phi}\approx\) 0.618 for \(t\to \infty\).
One can very well see that it is not a regular iteration at the left fixed point \(z_1\), because there is no interval of length \(>1\) such that \(f^{\circ t}\) is analytic at \(z_1\) for all t in this interval.
However I wonder why it seems to be regular at the right fixed point ...