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attracting fixed point lemma
#1
Attracting fixed point lemma, which for base would be:


where is a 1-cyclic function.

The more generalized lemma (obviously unproven) would be that for a given base B, RegularSuper_B(z), which is the entire regular super function, and which is developed from the primary repelling fixed point, and also given an analytic sexp_B(z), with singularities only at negative integers<=-2, sexp(-1)=0 and sexp(0)=1, sexp(z+1)=B^sexp(z), then


Where this gets somewhat interesting, is that for bases <= , sexp(z) is often developed from the attracting fixed point. For example, if B=, then the RegularSuper is entire, developed from the repelling fixed point of L=e at -infinity. And sexp(z) is developed from the attracting fixed point of L=e at +infinity. Which leads to my earlier claim for base eta. The claim is that sexp(z) for base eta exponentially converges to the SuperFunction(z+K) for base eta as imag(z) increases towards +I*infinity. K would be the first term in theta(z). Calculating theta(z) is equivalent to calculating the Kneser/Riemann mapping.

In the case for eta, decays to zero at +I*infinity. This is also the case for sexp(z) base e, where sexp(z) is usually developed from the Kneser/Riemann mapping.

A similar claim could be made for B=sqrt(2), where the two fixed points are L1=2, and L2=4. The regular super function is developed from the upper fixed point of L2=4. The claim would be that:


In this case, for sqrt(2), the period of the regular super function is 19.236, but the period of the sexp(z) is -17.143, so has singularities at n*17.143*I. At z=8.572*I, 1.047i+real valued 1-cyclic function. This real valued 1-cyclic function would be the wobble, where the two slightly differing functions go from f(z)=4 at -infinity, to f(z)=2 at +infinity.

I have started to calculated theta(z) for base sqrt(2), as well as for base eta. I wish I could generate Mike's beautiful .png graphics files! The leftmost contour shows the contour for eta, where the Imag(RegularSuper(z))=Pi*e*i, and real(RegularSuper(z)) goes from -infinity to +finity. maps this contour to sexp(z=-3) to sexp(z=-2). The next contour, z+1, has Imag(z)=0, and real(z) goes from -infity to 0. Mike's recent post http://math.eretrandre.org/tetrationforu...hp?tid=514 got me thinking that this equation might edit: or might not also hold for complex bases, which I hadn't considered earlier. Perhaps Mike's post is an example of a B=complex base, sexp(z) function, not developed from the fixed point, for which the might hold?

I will eventually calculate the theta(z) mapping for eta, and for sqrt(2), and post those numerical results. Again, here's the graph for the contour that needs to be mapped, for .
   
- Sheldon
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Messages In This Thread
attracting fixed point lemma - by sheldonison - 09/14/2010, 04:00 PM
new sexp(z) function for sqrt(2) - by sheldonison - 11/15/2010, 12:40 PM

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