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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 $\text{RegularSuper}_{\eta}(z)$.
[attachment=766]
- Sheldon
(09/14/2010, 04:00 PM)sheldonison Wrote: [ -> ]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.
.....
I will eventually calculate the theta(z) mapping for eta, and for sqrt(2), and post those numerical results....

$\text{newsexp}_{\sqrt(2)}(z)=\text{RegularSuper}_{\sqrt(2) \text{UpperL4}}(z+\theta(z))$

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 $\theta(z)$ 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.
[attachment=805]
[attachment=806]

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.
[attachment=807]
[attachment=808]

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.
[attachment=809]
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
(11/15/2010, 12:40 PM)sheldonison Wrote: [ -> ]... 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
Turns out it was just a simple typo, and in fact, it seems to converge very nicely. So, now I have calculated the 1-cyclic mapping for $\text{sexp}_{\sqrt 2}(z)=\text{usexp}_{\sqrt 2}(z+\theta(z)$. Below are the first ten terms linking the upper superfunction for the sqrt(2) with the sexp(z) for base the sqrt(2). In this post, I'm using the usexp notation from Henryk and Dimitrii's thread, although I don't normalize the usexp the same. Notice how quickly these fourier terms decay!
Code:
a0=5.284046911275929509562319765392910323367906148717443275834622148 a1=5.132787355776188711993056403737597090997350967306621139934333084E-27 -3.625724477536479451525596757218244963620871676800618582573812188E-25*I a2=1.511792461497588143794162537537493205516304722316568013968312358E-50 -6.197254624020296328476658293724870071698673149922065917475336508E-49*I a3=4.741105856335653880260054425878457273612939596030597725169056516E-74 -1.519169304975871323751503497236414682685357173501958294194139872E-72*I a4=1.570350059077682233462666140649351596259334241570698974766440939E-97 -4.321977227564835332037557803641919371360440249757521179671741019E-96*I a5=5.396718213010231485424958664652517410393596500561589465787103135E-121 -1.334210621916187903869149050260000544306712172666900701247539971E-119*I a6=1.903374565496422927307165800013735771632391677249739836118826118E-144 -4.336844598655851974923524833078588094464723019176568468495699952E-143*I a7=6.842567233214109482842955998942506385728260789623985911397625150E-168 -1.460695165744325250621260216290708048970792966625621627943202331E-166*I a8=2.496163638725303065423891681375328924239979050287271746885927788E-191 -5.049193271305711485904602482164353753575660842011093198286257629E-190*I a9=9.211945444661544659451549953304057311347259966944528834692313189E-215 -1.780248241034488000070344232012264715963937934250501299420384064E-213*I
$\text{sexp}(z+0.5*\text{period}_2) = \text{usexp (z+0.5*\text{period}_4+\theta(z))$

$\theta(z)=a_0 + \sum_{n=1}^\infty a_n\exp(2n\pi i*z) + \overline{a_n}\exp(-2n\pi i*z)$

This 1-cyclic theta(z) mapping is for the two nearly identical real valued functions, going from 4 at -infinity to 2 at +infinity. Here is a quick review, The usexp(z) function (upper superfunction developed from the repelling fixed point, L=4) is entire, with $\text{period}_4=19.236i = 2\pi i/\log(\log(4))$.
The sexp(z) function developed from the attracting lower fixed point, L=2, has a $\text{period}_2=17.143i = 2\pi i/\log(\log(2))$

At period4/2, usexp(z+period4/2), the upper superfunction, is real valued. At period2/2, sexp(z+period2/2) is also real valued. The two functions are nearly the same, except for a small wobble, discussed in this thread. What I calculated was the 1-cyclic $\theta(z)$ corresponding to that wobble! This $\theta$ is real valued at the real axis, and has singularities at +/-period2. The terms in the Fourier series correspond to terms in a Laurent series, with an annular ring of convergence.

This result actually came from another calculation, $\text{sexp}(z) = \text{usexp}(z+\theta(z))$, as part of showing that the fixed point lemma is probably true for b=sqrt(2). This is a different $\theta(z)$ function, where $a_n=b_n\exp(n\pi i*\text{period}_2)$.

$\theta(z)=b_0 + \sum_{n=1}^\infty b_n*\exp(2n\pi i*z) + \overline{b_n}*\exp(-2n\pi i*(\overline{z}+\text{period}_2))$
This $\theta(z)$ only converges when imag(z)>=0 and when imag(z)<=period2. $\theta(z)$ has singularities at the integer values of z at the real axis. Since sexp(z) is real valued at the real axis and at period2, the sexp(z) function can be analytically continued by schwarz reflection for imag(z)<0, or imag(z)>period2. Notice that the exp(-2nPi*i(z+period2)) terms are very small when z is near the real axis. I calculated the theta(z) mapping using a modified version of my kneser.gp. It is a similar, but different calculation than my previous post for the newsexp(z) function. The difference is that the iterated mapping needs to take into account the complex conjugate terms, exp(-2npiz) terms. Although, only the very first complex conjugate term is significant for results accurate to 64 decimal digits, but the presense of the conjugate terms changes the solution for all of the other terms as well. I felt I had to calculate results accurate to 64 decimal digits to distinguish this from the newsexp results I calculated in my previous post, since the two sets of theta(z) mappings agree to ~48 decimal digits of accuracy. To get results accurate to 64 decimal digits, I needed to calculate the first 192 terms of the 1-cyclic fourier series. This works if imag(z)>0.12*i. When imag(z)<0.12*i, the kneser.gp code uses the Taylor series calculated from a unit circle.

Here are the corresponding first 10 terms. Notice that these terms decay very slowly, due to the singularities at the real axis. I verified that the kneser mapping matches the sexp(z) developed from the attracting fixed point of L=2, to an accuracy of ~66 decimal digits. The other interesting thing, is that the kneser.gp mapping figured out the period2 of the sexp(z) function on its own, as a by product of calculating the kneser mapping. If you look at the b0 term below,$\text{period}_2=\text{period}_4-2i*\Im(b_0)$

Later, I will post graphs of how the theta(z) function grows towards the real contours of usexp(z), although I don't have much time right now since I'm going on vacation this weekend. I haven't gotten any feedback on this thread, so I hope someone else finds this interesting.
- Sheldon
Code:
b0=5.284046911275929509562319765392910323367906148717443275834622148 + 1.046500431344003802826235228141751177981429272641231666962572171*I b1=0.001259097259710786119123745778578435320167586725557454470469240301 -0.08894075358479671306467118402367967286235607271440657489260038903*I b2=0.0009097125541110377195626259677688333935732374987943475704079209011 -0.03729162881860908622115570317966209259425890769657591590319670327*I b3=0.0006998382430008637040089428220924149784163095581639022163612347651 -0.02242457370561371632560820434543429002085226680905505116659642705*I b4=0.0005686178651506032809589464446418805067258260073016471238767719519 -0.01564971740002252113273265858247721156519164240540210282152824495*I b5=0.0004793577727064269621919704364994948972034413404628596101373716670 -0.01185098437233873098569002007191946777135592682261501011791841248*I b6=0.0004147250799241742132243289924048833335081350136704092318540443246 -0.009449523259375778456636937567531642801956136930587550772560502650*I b7=0.0003657304476782259248553668197622578927606273056596151942251749570 -0.007807313814849530936408818127360955665585948908144433280127052591*I b8=0.0003272812333651051242946194223059395820860742622088032318313443531 -0.006620183771988586084865746743006685102250161066339740772096944563*I b9=0.0002962820999804195330669808904382806899677899478430886980151807773 -0.005725779538195315396861351431541602800561409436323770061855351858*I
Here is a contour plot for the $z+\theta(z)$ contour for $\text{usexp}_{\sqrt 2}(z)$, where usexp(0)~=5.767 (not normalized). The contour is $\theta(z)+z$ at the real axis, where $\text{sexp}_{\sqrt 2}(z)=\text{usexp}_{\sqrt 2}(z+\theta(z))$.
The first red contour represents the contour where $\Im(\text{usexp}(z))=\pi/\log(\sqrt 2)$.
This becomes sexp(z) between z=-3 and z=-2.
The next green contour represents $\text{usexp}(z)=-\infty \to 0$, This becomes sexp(z) between z=-2 and z=-1.
The second red contour represents $\text{usexp}(z)=0 \to 1$
The second green contour represents $\text{usexp}(z)=1 \to \sqrt 2$
- Sheldon
[attachment=812]
$\theta(z)+z$ has a similar contour at $\Im(z)=17.143$, exactly in between, at 8.571i, theta(z) is real valued, with a very small amplitude (4E-25). The "newsexp" z+theta(z) maps the same contour but without this second contour, and instead decays to 0 as imag(z) goes to infinity.
[attachment=814]
(11/15/2010, 12:40 PM)sheldonison Wrote: [ -> ]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 $\theta(z)$ decay to zero as imag(z) goes to +I*infinity.

This is really amazing. I guess I have again to become familiar with your way of computing what you call "Kneser mapping".