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On extension to "other" iteration roots
#1
This is a post discussing the possibility to generate other iterations, superfunctions, etc.

I. Classic Iteration Roots
It's very known that the eigendecomposition or fixed point method would work for any analytic functions having at least one constructable fixed point, which is precisely,


Having a deeper looking into the formula, and only considering the rational roots cases, which means t is rational, then the formula suddenly produces different iteration roots, as s^t has distinct branches as a multivalued function.
For example, , both satisfying the 1/2th iterational equation:, here s^{1/2} refers to the principle value, and we can see that the number of the iterative roots depend on the dominator of iteration index t.
This knowlegde allows us to combine different branches in the same "category" as MphLee called.

On the other hand, as we consider them as multivalued function, things get wierder. Still the case , we now define a multivalued function as one value mapped to a set, maybe infinite set, or .
The composition of two multivalued function defined as analogously to cartesian product, given f,g multivalued.
The second root gives, , we un-restrict the condition that, if one of the four values equals f(z), this multivalued function is called the second iterative root of f. The same thing applies to 3rd root, 4th root and so on.
Now remind the 2 different singlevalued branch cuts we used 
we then have exactly , where .
Intrinsically, we call g as another branch cut of f(z), because they are both the second iterative root to , which reads,
so we see that if all such functions are considered as multivalued functions, and we un-restrict the condition in that whenever there exists an element in equaling to another element in , the equation should be seen as "hold true",
then the multivalued functional iterations are self-consistent without nonconstructable fixed points.



II.  A generalization of the original fixed point method without theta mapping to construct a real-to-real superfunction?
The original fixed point method is to generate a series to approximate any iterations, in which we often use an asymptotic expansion of the superfunction: with a_n coefficients solvable.
And I've tested Sheldon's fatou.gp(really masterpiece), it's pitiful that the theta mapping of tetration based 1/2 would diverge. Also, since if we use the real fixed point , and call back to the definition of original tetration,  where , would yield a real-to-complex tetration since s<0. (You can tell theta mapping would lead to another real-to-complex map because the limit at aren't symmetric).
I'm inspired by another classic method, which is the piecewise approximation, to only find adequately good approximation between a strip with width 1 in the direction of Re(z), like , it can obtain real-to-real tetrations quickly, so also applicable of
The inspiration is that this method indicates the existence of a real-to-real tetration based 1/2, also it indeed gives beautiful results, however cannot be generalized to more general cases than tetration.
I'll call the method as "re-construction method", you may call it whatever you like, doesn't matter. The method already appeared once in my previous post.
Let's consider building up the real-to-real tetration base 1/2. We should notice that, if we denote   and , then they should behave asymptotically like:  at all integers n congruent to 0 modulo 2, since s is a negetive number. How about n congruent to 1 modulo 2? Simple.
will answer, due to the functional equation T satisfies.
Now we want a function that asymptotically behaves like T just at integers but also real-to-real, so that it preserves all integer values properly.
Consider a period 2 function which satisfies , then set a new "superfunction" , easily check that W preserves all integer values but different from T. 
Also we want W have the property real-to-real. Firstly, since s<0, , we can force that the map is real-to-real, as written and computed in the way: .
Then we only need to determine P(z), make it a real-to-real function is a striaghtforward solution, in my practical computation I used .
Now we call the function as our "initial guess", since P is not that known to precisely make for all real z. And then we use by our desire, to make W converge to a function, then the function is what we wanted.
I constructed the real-to-real tetration base 1/2 in this way, up to 10 decimal places correct(my computer works too slow!), indicating it should be valid for other cases, for any function, s<0, to construct a real-to-real superfunction.
This superfunction has even complex values, but has infinitely many branch cuts. I computed that it's limit at is that , showing an oscillative and indifferentiable behavior.
Finally we get results like (in a real-to-real sense) the value tables in the end of this post.

Update on 2021.9.25 10p.m.
To be noticed, this method can be seen as an approach different from theta mapping to merge two or more different superfunctions, the method can preserve each's value at integer points and then spawn a analytic solution.
To merge n superfunctions denoted as  with the same inter bases F^n (or for all i),  and preserve their integer values in a periodic sense:
As  denotes the merged version, its integer values are , and for all integers k is congruent to p modulo n,
The method to merge them is then called, aforementioned "re-construction method" or anything you prefer.
To do so, we introduce a periodic function P with period n, .
To construct an initial guess  and preserve all integer values in the desired way, let , which can be achieved by modifying the coefficients in the fourier series of P, and there's already a well-known such P, , to only generate a real-to-real P, just consider a cosine fourier series of P is sufficient.
And set
Then we just simply use to make it converge. And we see if W converges, it is a superfunction of , so we simply take . Then all integer values of the n superfunctions is settled on all multiples of  ordered in our desired way.

Update on 2021.9.25 11p.m.
When using the method II., it's very useful of these statements to test if P is proper enough to make W converge:
1. Whenever there lies a point  in the nonintegers, and it lies within the convergence of , the P function won't work. Since all neighborhood of z_0 will converge to the same fixed point after sufficiently high order of iteration, unless all T have the same limit at infinity.
For example, consider to construct a merged tetration based 0.06, from 2 superfunctions . T1 takes its value above the fixed point L~0.54323 of F, T2 takes its value below the fixed point L2~0.36158, if we choose , suddenly we get , which lies within the convergence (will converge to ~=0.36158 which is L2), so this P function cannot work.
Also, it shows that such P won't work for the case F(z)=-z(1-z).
2. From 1, we instantly know that if such cases exists, we must choose a P function having similar behavior to secant function, its range will never reach within some interval.
It suddenly derives that for tetration bases 0<b<e^-e, there is no real-to-real tetration, since it bounces up and down between 2 different fixed points' neighborhood, somewhere in the oscillation will be involved into the convergence of its fixed point, and therefore almost all real values will get nearby to only one fixed point, and the other repells all values during iteration, so whether it is a indifferentiable functions only take its original integer values at all real numbers, or it doesn't exist. So the bases 0<b<e^-e are remaining unsolved if we want a merged version, as far as I know. (Theta mapping will diverge for these bases.)
3. If all T have the same limit at infinity, or to say, they are constructed from the same fixed point, then almost any P will work.
You can see this applies to the tetration based 0.5, since we chose two branches that all converge to ~0.6. We can apply this to create a tetration based sqrt(2) oscillates between 1 and 3 when z>1. I'll do this sooner, though.

I'll put this later in my paper maybe.
This post may be updated.
I'm a freshman to university and the homework is far more than I expected, so I have little time to do research, so saaaad.

Attached files:
p1: the real-to-real tetration base 1/2 along the real axis

p2: the real-to-real tetration base 1/2 on the imaginary axis

Real Value Table

Code:
{0.00, 1.0000000000},
{0.01, 0.9966247042},
{0.02, 0.9929029003},
{0.03, 0.9888461961},
{0.04, 0.9844664906},
{0.05, 0.9797759373},
{0.06, 0.9747869084},
{0.07, 0.9695119587},
{0.08, 0.9639637914},
{0.09, 0.9581552245},
{0.10, 0.9520991582},
{0.11, 0.9458085436},
{0.12, 0.9392963527},
{0.13, 0.9325755499},
{0.14, 0.9256590649},
{0.15, 0.9185597669},
{0.16, 0.9112904407},
{0.17, 0.9038637639},
{0.18, 0.8962922860},
{0.19, 0.8885884088},
{0.20, 0.8807643686},
{0.21, 0.8728322195},
{0.22, 0.8648038184},
{0.23, 0.8566908118},
{0.24, 0.8485046230},
{0.25, 0.8402564415},
{0.26, 0.8319572136},
{0.27, 0.8236176337},
{0.28, 0.8152481370},
{0.29, 0.8068588936},
{0.30, 0.7984598033},
{0.31, 0.7900604914},
{0.32, 0.7816703053},
{0.33, 0.7732983125},
{0.34, 0.7649532983},
{0.35, 0.7566437656},
{0.36, 0.7483779340},
{0.37, 0.7401637405},
{0.38, 0.7320088404},
{0.39, 0.7239206082},
{0.40, 0.7159061400},
{0.41, 0.7079722556},
{0.42, 0.7001255005},
{0.43, 0.6923721497},
{0.44, 0.6847182099},
{0.45, 0.6771694239},
{0.46, 0.6697312732},
{0.47, 0.6624089825},
{0.48, 0.6552075233},
{0.49, 0.6481316182},
{0.50, 0.6411857445},
{0.51, 0.6343741392},
{0.52, 0.6277008025},
{0.53, 0.6211695028},
{0.54, 0.6147837804},
{0.55, 0.6085469523},
{0.56, 0.6024621164},
{0.57, 0.5965321558},
{0.58, 0.5907597430},
{0.59, 0.5851473446},
{0.60, 0.5796972248},
{0.61, 0.5744114504},
{0.62, 0.5692918943},
{0.63, 0.5643402401},
{0.64, 0.5595579856},
{0.65, 0.5549464473},
{0.66, 0.5505067640},
{0.67, 0.5462399008},
{0.68, 0.5421466527},
{0.69, 0.5382276488},
{0.70, 0.5344833555},
{0.71, 0.5309140802},
{0.72, 0.5275199751},
{0.73, 0.5243010407},
{0.74, 0.5212571289},
{0.75, 0.5183879468},
{0.76, 0.5156930599},
{0.77, 0.5131718954},
{0.78, 0.5108237454},
{0.79, 0.5086477704},
{0.80, 0.5066430020},
{0.81, 0.5048083465},
{0.82, 0.5031425878},
{0.83, 0.5016443902},
{0.84, 0.5003123021},
{0.85, 0.4991447584},
{0.86, 0.4981400837},
{0.87, 0.4972964951},
{0.88, 0.4966121056},
{0.89, 0.4960849263},
{0.90, 0.4957128698},
{0.91, 0.4954937530},
{0.92, 0.4954252998},
{0.93, 0.4955051439},
{0.94, 0.4957308318},
{0.95, 0.4960998257},
{0.96, 0.4966095061},
{0.97, 0.4972571744},
{0.98, 0.4980400563},
{0.99, 0.4989553042},
{1.00, 0.5000000000}

Imaginary value table
Code:
{0. + 0.00 I, 1.0000000000 + 0.0000000000 I},
 {0. + 0.01 I, 1.0001789242 - 0.0032001040 I},
 {0. + 0.02 I, 1.0007158797 - 0.0064113234 I},
 {0. + 0.03 I, 1.0016114147 - 0.0096448108 I},
 {0. + 0.04 I, 1.0028664430 - 0.0129117937 I},
 {0. + 0.05 I, 1.0044822442 - 0.0162236121 I},
 {0. + 0.06 I, 1.0064604638 - 0.0195917577 I},
 {0. + 0.07 I, 1.0088031133 - 0.0230279128 I},
 {0. + 0.08 I, 1.0115125700 - 0.0265439912 I},
 {0. + 0.09 I, 1.0145915774 - 0.0301521799 I},
 {0. + 0.10 I, 1.0180432442 - 0.0338649833 I},
 {0. + 0.11 I, 1.0218710443 - 0.0376952678 I},
 {0. + 0.12 I, 1.0260788153 - 0.0416563103 I},
 {0. + 0.13 I, 1.0306707570 - 0.0457618477 I},
 {0. + 0.14 I, 1.0356514289 - 0.0500261303 I},
 {0. + 0.15 I, 1.0410257469 - 0.0544639774 I},
 {0. + 0.16 I, 1.0467989785 - 0.0590908374 I},
 {0. + 0.17 I, 1.0529767371 - 0.0639228510 I},
 {0. + 0.18 I, 1.0595649740 - 0.0689769190 I},
 {0. + 0.19 I, 1.0665699686 - 0.0742707750 I},
 {0. + 0.20 I, 1.0739983153 - 0.0798230636 I},
 {0. + 0.21 I, 1.0818569081 - 0.0856534237 I},
 {0. + 0.22 I, 1.0901529208 - 0.0917825789 I},
 {0. + 0.23 I, 1.0988937832 - 0.0982324344 I},
 {0. + 0.24 I, 1.1080871509 - 0.1050261821 I},
 {0. + 0.25 I, 1.1177408697 - 0.1121884133 I},
 {0. + 0.26 I, 1.1278629321 - 0.1197452409 I},
 {0. + 0.27 I, 1.1384614244 - 0.1277244310 I},
 {0. + 0.28 I, 1.1495444632 - 0.1361555454 I},
 {0. + 0.29 I, 1.1611201200 - 0.1450700948 I},
 {0. + 0.30 I, 1.1731963296 - 0.1545017044 I},
 {0. + 0.31 I, 1.1857807825 - 0.1644862915 I},
 {0. + 0.32 I, 1.1988807949 - 0.1750622570 I},
 {0. + 0.33 I, 1.2125031561 - 0.1862706896 I},
 {0. + 0.34 I, 1.2266539463 - 0.1981555846 I},
 {0. + 0.35 I, 1.2413383210 - 0.2107640750 I},
 {0. + 0.36 I, 1.2565602570 - 0.2241466773 I},
 {0. + 0.37 I, 1.2723222519 - 0.2383575469 I},
 {0. + 0.38 I, 1.2886249704 - 0.2534547447 I},
 {0. + 0.39 I, 1.3054668294 - 0.2695005093 I},
 {0. + 0.40 I, 1.3228435103 - 0.2865615308 I},
 {0. + 0.41 I, 1.3407473895 - 0.3047092200 I},
 {0. + 0.42 I, 1.3591668735 - 0.3240199626 I},
 {0. + 0.43 I, 1.378085627 - 0.344575348 I},
 {0. + 0.44 I, 1.397481677 - 0.366462355 I},
 {0. + 0.45 I, 1.417326384 - 0.389773472 I},
 {0. + 0.46 I, 1.437583259 - 0.414606731 I},
 {0. + 0.47 I, 1.458206621 - 0.441065612 I},
 {0. + 0.48 I, 1.479140084 - 0.469258774 I},
 {0. + 0.49 I, 1.500314861 - 0.499299581 I},
 {0. + 0.50 I, 1.521647910 - 0.531305322 I},
 {0. + 0.51 I, 1.543039919 - 0.565396089 I},
 {0. + 0.52 I, 1.564373184 - 0.601693195 I},
 {0. + 0.53 I, 1.585509428 - 0.640317064 I},
 {0. + 0.54 I, 1.606287674 - 0.681384480 I},
 {0. + 0.55 I, 1.626522293 - 0.725005100 I},
 {0. + 0.56 I, 1.646001420 - 0.771277167 I},
 {0. + 0.57 I, 1.664485977 - 0.820282377 I},
 {0. + 0.58 I, 1.681709594 - 0.872079909 I},
 {0. + 0.59 I, 1.697379756 - 0.926699745 I},
 {0. + 0.60 I, 1.711180542 - 0.984135480 I},
 {0. + 0.61 I, 1.722777277 - 1.044337017 I},
 {0. + 0.62 I, 1.731823358 - 1.107203660 I},
 {0. + 0.63 I, 1.737969359 - 1.172578293 I},
 {0. + 0.64 I, 1.740874283 - 1.240243436 I},
 {0. + 0.65 I, 1.740218537 - 1.309919984 I},
 {0. + 0.66 I, 1.735717876 - 1.381269344 I},
 {0. + 0.67 I, 1.727137241 - 1.453899405 I},
 {0. + 0.68 I, 1.714303246 - 1.527374429 I},
 {0. + 0.69 I, 1.697114000 - 1.601228413 I},
 {0. + 0.70 I, 1.675545158 - 1.674981012 I},
 {0. + 0.71 I, 1.649651475 - 1.748154708 I},
 {0. + 0.72 I, 1.619563672 - 1.820291725 I},
 {0. + 0.73 I, 1.585481005 - 1.890969196 I},
 {0. + 0.74 I, 1.547660399 - 1.959811414 I},
 {0. + 0.75 I, 1.506403378 - 2.026498391 I},
 {0. + 0.76 I, 1.462042062 - 2.090770488 I},
 {0. + 0.77 I, 1.414925480 - 2.152429309 I},
 {0. + 0.78 I, 1.365407136 - 2.211335433 I},
 {0. + 0.79 I, 1.313834490 - 2.267403730 I},
 {0. + 0.80 I, 1.260540651 - 2.320597086 I},
 {0. + 0.81 I, 1.205838351 - 2.370919287 I},
 {0. + 0.82 I, 1.150016003 - 2.418407711 I},
 {0. + 0.83 I, 1.093335582 - 2.463126277 I},
 {0. + 0.84 I, 1.036031960 - 2.505159003 I},
 {0. + 0.85 I, 0.978313358 - 2.544604321 I},
 {0. + 0.86 I, 0.920362594 - 2.581570250 I},
 {0. + 0.87 I, 0.862338844 - 2.616170399 I},
 {0. + 0.88 I, 0.804379702 - 2.648520767 I},
 {0. + 0.89 I, 0.746603372 - 2.678737239 I},
 {0. + 0.90 I, 0.689110864 - 2.706933693 I},
 {0. + 0.91 I, 0.631988119 - 2.733220618 I},
 {0. + 0.92 I, 0.575307999 - 2.757704147 I},
 {0. + 0.93 I, 0.519132119 - 2.780485441 I},
 {0. + 0.94 I, 0.463512500 - 2.801660326 I},
 {0. + 0.95 I, 0.408493054 - 2.821319149 I},
 {0. + 0.96 I, 0.354110881 - 2.839546792 I},
 {0. + 0.97 I, 0.300397419 - 2.856422800 I},
 {0. + 0.98 I, 0.247379429 - 2.872021604 I},
 {0. + 0.99 I, 0.195079845 - 2.886412804 I},
 {0. + 1.00 I, 0.143518502 - 2.899661495 I}


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On extension to "other" iteration roots - by Leo.W - 09/24/2021, 04:25 PM

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