06/04/2011, 03:05 PM
(This post was last modified: 06/04/2011, 03:17 PM by sheldonison.)
(06/04/2011, 02:10 PM)bo198214 Wrote:Yeah, I think that's a big change in the function there. Do you have a reference on the "shell tron region"? I seem to remember reading about it many times, "Kidney bean shaped" region or something, but I don't know what the theory tells us happens.(06/04/2011, 01:56 PM)sheldonison Wrote: I calculated what happens to the fixed points, in going from circle base \( =2\eta+\sqrt{2} \) to base\( =\sqrt{2} \). Moving along the lower circular path, "underneath" eta, I can seamless morph the upper repelling fixed point 2.478+0.8518i seamlessly to a repelling fixed point of 4.0. But the lower fixed point, which starts out at 2.478-0.8518i, and eventually becomes a fixed point of 2.0, somewhere early along the path, around 7 degees/180, the repelling fixed point of 2.478-0.8518i becomes an attracting fixed point of 2.435 - 0.8239i, before continuing on as an attracting fixed point, moving towards 2.0
The switch repelling/attracting imho takes place on the Shell-Tron-boundary.
Also, Kneser mappings probably won't work from attracting fixed points, unless there are repeating regions of superexponential growth as real(x) grows. For example, the lower sexp superfunction of eta definitely lacks such superexponential growth, as real(x) grows. Also, the lower superfunction for sexp sqrt(2). Instead, both of those functions converge to the fixed point as real(x) grows, for all imag(z).
- Sheldon