11/15/2010, 12:40 PM
(This post was last modified: 11/15/2010, 02:14 PM by sheldonison.)

(09/14/2010, 04:00 PM)sheldonison Wrote: Attracting fixed point lemma, which for base would be:

where is a 1-cyclic function.

.....

I will eventually calculate the theta(z) mapping for eta, and for sqrt(2), and post those numerical results....

Well, I have results for sqrt(2), and, its a different sexp(z) base sqrt(2) function than any of the four functions described in Henryk and Dimitrii's paper! The new sexp(z) function was calculated as a 1-cyclic mapping of the regular superfunction developed from the repelling upper fixed point of 4, where all of the terms of the decay to zero as imag(z) goes to +I*infinity.

At the real axis, the new sexp(z) behaves almost exactly like the sexp(z) developed from the attracting fixed point of 2. sexp(-1)=0, sexp(0)=1, sexp(1)=sqrt(2). There are singularities at z=-2,-3,-4.... integers to minus infinity, and it is real valued at the real axis for z>2. However, the function is pseudo periodic, with a pseudo periodicity of 19.236*I, matching the periodicity of the entire superfunction developed from the repelling fixed point!

Unlike the function described by Henryk and Dimitrii, all of the singularities are at the real axis. Also, because the new sexp(z) is pseudo periodic, as oppossed to periodic, there is only one horizontal line at imag(z)=0, for which imag(sexp(z))=0. I believe that perhaps what I am describing is possibly the Perterbed Fatou solution, described in Henryk's post.

The first graph shows this new sexp(z) at the real axis, with its limiting behavior of 2 as x goes to infinity. And the next graph shows the very very tiny difference between the new sexp(z) function and the sexp(z) developed from the attracting fixed point of L=2, graphed at the real axis. The two functions are identical, to nearly 48 decimal digits of accuracy! At x=0.25, newsexp(x)-sexpL2(x)=1.4*10^-48.

The next two graphs show the behavior of the new sexp(z) function at imag(z)=8.571*I, which is half the periodicity of the attracting fixed point. Real(z) varies from -10 to +4. Here the new sexp(z) is nearly real valued. The second graph shows the imag(z) not being real valued (green contour), while the smaller red value shows the real difference between the new sexp(z), and the sexp(z) developed from the attracting fixed point of 2. For comparison, the 2x larger red value shows the real difference wobble described by Khouznetsov and Trappman's paper.

At larger values of imag(z), the new sexp(z) quickly decays towards the regular entire superfunction. Here, imag(z)=18.19*i, imag(sexp(z)), is around 3*10^-51, and the function shows superexponential growth.

The results were calculated with an updated version of my kneser.gp code, where I fixed the initialization code, so it works for bases<eta; the older kneser.gp version hangs for bases<eta. The mapping was computed accurate to approximately 64 decimal digits. I also tried to generate the mapping from sexp (repelling,L=4) to sexp(attracting,L=2), but the algorithm I'm using in kneser.gp only converges if the theta(z) value decays to zero as imag(z) goes to +i*infinity. However, the mapping from L=4 to L=2 is very close to the mapping I generated.

- Sheldon