new results from complex dynamics

[mike3]

edited Where both fixed points are repelling, there is only one solution, and the fixed points cannot be swapped (check your math, and/or else post of sexp(z)). Henryk has a paper with a uniquess criteria for sexp(z), and I hope to eventually show the theta(z) uniqueness criteria I posted yesterday, \( \theta(z)=\text{slog}(f(z))-z \), can be shown to be complete if both fixed points are repelling and theta(z) has a singularity.

So I guess I was right -- the strange fixpoint-reversal phenomenon probably simply means the continuation is impossible.

But, inside the Shell Thron boundary, where one of the fixed points is attracting, there are two different solutions for the same base, which also makes the uniqueness criteria more complicated. Are you calculating sexp(z), or slog(z)? I've started to make plots of slog(z) as well, for the bipolar solutions, inside the Shell Thron boundary. Also, as you point out, as you approach the second Shell Thron crossing, the singularities for sexp(z) in the upper half of the complex plane (rotating around eta counterclockwise) start to bunch up and get arbitrarily close to each other and the real axis.
- Sheldon

I was calculating the fixed points to see how the sickel would behave at the boundary crossing, to see if it "exploded" or something weird like that.

How do you graph slog(z), anyway -- i.e. how do you handle all the branch cutting and so forth?