[Update] Comparision of 5 methods of interpolation to continuous tetration

[sheldonison]

The only thing I can come up with is that :

1) we simply take the principal schroeder function near the fixpoint c.

2) in that domain we use analytic continuation ( monodromy ) so that by small radiuses recentered we can reach any value in the upper plane.

3) In that upper plane we find ( after many continuations ) the values equalling the reals between 0 and 1. ( and those are connected )
( for reasons yet to be explained !)
.....
10) we have sexp analytic near the positive real line.
....
And if I got It right , there is still alot to be explained !!

Tommy,

Is your question about using Kneser's method to get numerical results, or is it about Kneser's proof of a real valued sexp(z) solution? It is not practical to use Kneser's method to generate usable results for sexp(z), though it is a proof of the existence of such a real valued solution, even though numerically, Kneser's Riemann mapping is very difficult to work with. Also, Kneser's proof is hard to follow, and I am not good at math to recreate it.

On the other hand, I wrote a pari-gp program that is numerically equivalent to Kneser's method that is practical, though I cannot rigorously prove my method converges. But assuming my method converges, than I can show it converges to the same solution as would be given by Kneser's Riemann mapping solution. Obviously, I could explain my method, and its equivalence to Kneser's solution, but that would not be relevant to your questions.
- Sheldon

[tommy1729]

I simply asked if I am correct with my 10 step way to do kneser.

I do not ask for the Riemann mapping so I guess its more of a proof question.

But actually I'd say its a " construction " question.

Is the Kneser proof/solution constructed in the 10 steps I posted or is one or more steps wrong ?

I assumed sheldon was an expert on kneser.

Maybe bo is.

I must say Im not so good with code.

And actually I wonder , if you cannot explain kneser completely , how can you find your own method based on kneser and equivalent to kneser ??

That is a mystery to me.

Also apart from the fact that i can read only math - not so good with code - I also wonder how Im suppose to fully understand an equivalent method if I do not understand the Original completely ?!

ON the other hand , Im convinved that once I understand Kneser , I will be able to prove convergeance of your method ....

Maybe Bo or Jay can help us out here.

Regards

tommy1729

[sheldonison]

I simply asked if I am correct with my 10 step way to do kneser.

I do not ask for the Riemann mapping so I guess its more of a proof question.

But actually I'd say its a " construction " question.

Is the Kneser proof/solution constructed in the 10 steps I posted or is one or more steps wrong ?

Tommy, the sequence is good, but the Riemann mapping step 5) has a lot of sub-steps. For me, the biggest complexity hurdle in Kneser's construction, besides the fact that I don't have a formal math degree, is taking the \( \alpha(z) \) Abel function, from Schroeder function, of the real axis, which is after the step where he generates the chi-star, but still one or two steps before the Riemann mapping. I don't read German, so have no idea how he proved the infinite region is simply connected, and I wouldn't know how to do so, since the region is increasingly recursively complex.

Here is the rough Abel function of the real axis, showing the repeating pattern; here \( \Im(z)=\frac{1}{sexp(3.5)}\approx 10^{-78} \). Kneser multiplies this repeating pattern by \( 2\pi i \) and then takes the exponent of that; \( \exp(2\pi i z) \). That is the contour that gets wrapped around a unit circle for the Riemann mapping.
   

Here we zoom in on one of the singularities, where sexp(z)=0. The singularity gets ever more complicated as we super-exponentially approach zero. Here, I show what the contour looks like if \( \Im(z)=\frac{1}{\text{sexp}(8.5)} \).
   

My algorithm has a mathematical description, as well as pari-gp code. I don't want to side track too much, but it does something different but equivalent to generate \( \text{sexp}(z)=\alpha^{-1}(z+\theta(z)) \), via a 1-cyclic mapping from the inverse Abel function, \( \alpha^{-1}(z) \), as well as an sexp(z) Taylor series representation at the real axis. This is because the \( \theta(z) \) function has a singularity at the real axis, so adequate convergence is not possible with a reasonable number of terms. So my algorithm actually has to iteratively generate two different equivalent representations of sexp(z).
- Sheldon

[tommy1729]

Thank you sheldon.

At this point I wonder most about how the riemann mapping of the abel function maintains its functional equation.

I recall knowing this but I forgot ... if I recall well. ( how much doubt about memory can one sentence have ? :p )

Regards

tommy1729

[Gottfried]

An application of the "polynomial method/Diagonalization" (matrix-size 64x64) which was in the instance of the article shown with base \( b=4 \) and in that article likely asymptotic to the Kneser method.

Here I provide a picture with focus on complex iteration from one real starting point \( z_0 = 1 \) with base \( b=1.3 \). (I've not yet checked against Sheldon's Kneser-implementation ).

The picture shows roughly circles: along the circumferences the iteration-height is purely imaginary; one revolving means \( 2 \pi i / v \) where v is the log of the log of the fixpoint \( t \approx 1.48 \)


   

It is still surprising to me that we can proceed from one real point below the fixpoint to some other real point above the fixpoint - which means to avoid/surpass the infinite height-iteration: just by using imaginary heights...


Gottfried

Here I added two more (hopefully instructive) views;

Here the base-point is z0=0
   

and here is the iteration to one more negative height, where I had to leave out the infinitely distant point z0 (-> - infinity)

   

Now I've got my matrices for base b=1.44, near eta. What a mess!
I don't have any idea - there is no obvious divergence in the power series with 64 terms. I also took z0=b^b ~ 1.69 as initial point; the curves originating from z0=1 were even more messed up.
   

This is the picture base b=1.44 z0=1+0î - I've no explanation so far for the messed curves.

   


<hr>


P.s.: A 3-D picture with colors indicating height, and "isobares"-grid and the fixpoint shown as peak of infinite height were nicer but I do not know how to draw one(which software). If someone else likes to play with this I can provide the coefficients of the diagonalization matrices in Pari/GP-convention: the computation of that matrices is extremely costly (it needed 6000 secs to be computed and more than 3000 digits decimal precision;I chose then 4000 digits) so it might be interesting to get the ready-made numbers by download instead by a new computation.

[tommy1729]

Very nice pic Gottfried.

I think the phenomenon is not typical for tetration but for most functions with a limited amount of fixpoints.

What surprises me is that the shapes are so close to perfect circles.

I wonder how it looks like if the real fixpoints approaches its limits.

In other words what happens if the base gets close to eta ?

Do we get figure 8 shapes instead of circles because of the anticipation of the pair of conjugate fixpoints ?

What happens with functions with 2 or more fixpoints ? do they also have these circles around their fixpoints ? In other words, do they locally ( around the fixpoints ) behave the same as the plot given by gottfried here ?

Why not ellipses ??

And what if there is no fixpoint ?

Maybe we need riemann surfaces to understand this better ?

Many questions as usual.

Regards

Tommy1729

[Gottfried]

Very nice pic Gottfried.

I wonder how it looks like if the real fixpoints approaches its limits.

In other words what happens if the base gets close to eta ?

Hmm, I'm trying hard to get pictures with such bases, but for some reason (which I do not yet understand and thus cannot compensate for), this becomes extremely difficult - for instance: perhaps I'll need power series with thousands of terms, I don't know. I'm still fiddling with it to find the key to access the problem in a reasonable generality/range of bases - so this might take some time...

Gottfried

[Gottfried]

What surprises me is that the shapes are so close to perfect circles.

I wonder how it looks like if the real fixpoints approaches its limits.

In other words what happens if the base gets close to eta ?

Do we get figure 8 shapes instead of circles because of the anticipation of the pair of conjugate fixpoints ?

What happens with functions with 2 or more fixpoints ? do they also have these circles around their fixpoints ? In other words, do they locally ( around the fixpoints ) behave the same as the plot given by gottfried here ?

Why not ellipses ??

Concerning the shape:[please see my updated previous post, I've inserted two new pictures] as the base point \( z_0=1 \) goes to the left then the shape becomes first a distorted oval (picture 2) and later a -left-open ellipsoidic looking shape going to negative infinity to the left (picture 3).

Note also on the real axis, righthand of the fixpoint, the occurence of some "mirror"- or "Reflexion"-points: just for the two additional iterations to the right(=negative height) from the currrently most right point ( \( \text{reflexion}(z_0) = y_0 \approx 1.86300282587 \) the "zero-reflexion" \( \text{reflexion} (z_{-1}) = y_{-1} \approx 2.37147238675802 \) and then "neg infinity-reflexion" \( \text{reflexion}(z_{-2})= y_{-2} \approx 3.29126767409758 \)).

The left part of the curve has then the horizontal lines +-<not-yet-determined-imaginary-value> as limiting asymptote.

Well, I'm really curious too what the shape is with bases nearer to eta...

Gottfried

[tommy1729]

Note also on the real axis, righthand of the fixpoint, the occurence of some "mirror"- or "Reflexion"-points: just for the two additional iterations to the right(=negative height) from the currrently most right point ( \( \text{reflexion}(z_0) = y_0 \approx 1.86300282587 \) the "zero-reflexion" \( \text{reflexion} (z_{-1}) = y_{-1} \approx 2.37147238675802 \) and then "neg infinity-reflexion" \( \text{reflexion}(z_{-2})= y_{-2} \approx 3.29126767409758 \)).

The left part of the curve has then the horizontal lines +-<not-yet-determined-imaginary-value> as limiting asymptote.

Gottfried

Im not sure what you meant by reflexion. I noticed the fixpoint is a red x in your plots and there are also green x's. I guess that relates.
Maybe its about pseudocenters of the pseudocircles or the analogue for ellipses ( pseudofocal points) . It seems these centers move away from the fixpoints. These centers seem to follow the real iterations although probably not linearly.

The idea of circles becoming other shapes gradually , reminds me of pseudoperiodic functions. Afterall its like f(a+2pi i) = f(a) but f(a+1+2pi i) =/= f(a+1). Kinda.

As for the asymptotic to the left I wonder - based on the idea of pseudoperiodic somewhat too - if that +-<not-yet-determined-imaginary-value> is actually 1.3 2 pi i ? From the formula b 2pi i where b is the base for 1<<b<<eta.

Perhaps a bit naive. I need more training.

As for these shapes *very* near the fixpoints I believe they are mainly determined by the first 2 derivatives of the fixpoint.

The shapes *far* away are then just the ln or exp iterations of the others (ln and exp in base b of course) . This seems to make sense.

For instance *near* the fixpoints if the function is well approximated by ( recenter fixpoint at 0 for simplicity ) :

0 + x + (x^2)/15

,then we get a circle approximation since the derivative is 1.

This is because the complex iterations of x around x=0 are also a circle.

Maybe a bit naive but my first impression.

Thanks for the updates.

regards

tommy1729

[Gottfried]

Im not sure what you meant by reflexion. I noticed the fixpoint is a red x in your plots and there are also green x's. I guess that relates.

The green x is the position of the base. With "reflexion" I hope to introduce a meaningful expression for the impression, that the two points on the opposite sides of the "circle" - opposite with respect to the fixpoint- look like mirrored images of each other - and are also just looking like half-circle rotations of each other (which would then be a very nice and significant property).

With the <not-yet-determined-imaginary-value> yes, that 1.3*2*pi is perhaps a good idea. I think it should have a simple solution; I just don't get it at the moment... [update] It should be 2*Pi*I/v where v= log(log(LH(1.3))) where the function t=LH(b) gives the fixpoint t such that b^t=t and v is about 1/v~-1.0503


With the other questions - I do not yet know...



Gottfried