09/14/2010, 04:00 PM
(This post was last modified: 09/14/2010, 05:54 PM by sheldonison.)
Attracting fixed point lemma, which for base \( \eta \) would be:
\( \text{sexp}_{\eta}(z)=\text{RegularSuper}_{\eta}(z+\theta(z)) \)
where \( \theta(z) \) 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
\( \text{sexp}_B(z) = \text{RegularSuper}_B(z+\theta(z)) \)
Where this gets somewhat interesting, is that for bases <= \( \eta \), sexp(z) is often developed from the attracting fixed point. For example, if B=\( \eta \), 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, \( \theta(z) \) 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:
\( \text{sexp}_{\sqrt(2)}(z)=\text{RegularSuper}_{\sqrt(2)}(z+\theta(z)) \)
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 \( \theta(z) \) has singularities at n*17.143*I. At z=8.572*I, \( \theta(z)\approx \)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. \( \theta(z) \) 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 \( \theta(z) \) 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 \( \theta(z) \) mapped, for \( \text{RegularSuper}_{\eta}(z) \).
- Sheldon
\( \text{sexp}_{\eta}(z)=\text{RegularSuper}_{\eta}(z+\theta(z)) \)
where \( \theta(z) \) 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
\( \text{sexp}_B(z) = \text{RegularSuper}_B(z+\theta(z)) \)
Where this gets somewhat interesting, is that for bases <= \( \eta \), sexp(z) is often developed from the attracting fixed point. For example, if B=\( \eta \), 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, \( \theta(z) \) 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:
\( \text{sexp}_{\sqrt(2)}(z)=\text{RegularSuper}_{\sqrt(2)}(z+\theta(z)) \)
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 \( \theta(z) \) has singularities at n*17.143*I. At z=8.572*I, \( \theta(z)\approx \)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. \( \theta(z) \) 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 \( \theta(z) \) 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 \( \theta(z) \) mapped, for \( \text{RegularSuper}_{\eta}(z) \).
- Sheldon
