new results from complex dynamics

[sheldonison]

On the other hand if I remember, Sheldon posted somewhere that he also found a solution for the non-primary fixpoints. The secondary fixpoints however do not collapse into a horizontal fixpoint pair, but they remain vertical when passing \( b=e^{1/e} \) on the real axis. Though I really wonder whether there can exist a sickle between these fixpoints, i.e. an area bounded by an (injective) curve between both fixpoints and its image under \( b^z \).

http://math.eretrandre.org/tetrationforu...hp?tid=452
There's a lot of good posts here, but one quick comment on the non-primary fixed point solution which I posted. The solution from the alternative fixed point doesn't have an analytic abel function at real axis! That's because the first and second derivatives of that sexp(z) goes to zero at every integer>=-2, which leads to singularities for the abel function for sexp(n) where the derivative gets infinitely large at the real axis. So theorems about the Fatou coordinate (or Abel function) wouldn't apply to this weird alternative fixed point solution. Only the solution from the primary fixed point has an analytic abel function at the real axis, with the derivative of the sexp(z) function>0 for all z>-2.
- Sheldon