attracting fixed point lemma

[sheldonison]

... 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