[AIS] (alternating) Iteration series: Half-iterate using the AIS?

[sheldonison]

Hi Sheldon -

... Then the empirical observation that (it) is sort of "hybrid", ... and the fractional iteration based on it should then as well be taken as "hybrid" of the developments at the two different fixpoints (where one should explicite the concrete meaning of the "hybrid"ity into a formalized algebraic description).

Thanks Gottfried. I think its an interesting summation, and probably leads to an analytic abel function via, \( \alpha(z)=\frac{1}{\pi}\sin^{-1}(\frac{\text{asum}(z)}{\text{amplitude}}) \). My hunch is that the superfunction, \( \alpha^{-1}(z) \), would have fractal singularities, as imag(z) increases approaching the Period/2 for the lower fixed point, but that is only a pure intuitive hunch, not based on any data or numerical results.
- Sheldon