Kneser-iteration on n-periodic-points (base say \sqrt(2))

[JmsNxn]

Hi James -

yes, you've got correctly what I meant with the logarithmizing and the "key".                    

But then you formulate

However, the Kneser function knows to do this; such that the map \( \log \) will necessarily choose the right branch (or rather, the right sequence of branches).

Hmm, I don't know how I should understand this...

......

update
After some more working with the data as described, I believe I have a proof, that a continuous curve, connecting all the three periodic points \(p1,p2,p3,p1,...\) in the sense of representing the trajectory of continuous iteration, cannot exist:       
If there is such a curve, then each point on one partial curve connecting a pair of points (for instance \(p1,p2\)), must as well be 3-periodic. But this would mean the number of 3-periodic points would be uncountably infinite (and for this single example of a 3-period only). But it has been proved elsewhere, that the whole number of n-periodic points for each n is only countably infinite.
/end update


Kind regards-
Gottfried

Hey, Gottfried!

Your update is much more what I was driving at. If we take a key, and talk about producing an iteration;

\(
\log_k(z_k) = z_{k-1}\\
\)

And drawing a spline between them--then it must disagree with Kneser at some point, because Kneser eventually tends towards the value \( L \) as we iterate it. Which is, it no longer satisfies the periodic structure. So, as you are proving this by saying it produces uncountable periodic points; I'm saying something slightly different.

That, for an appropriate sequence of logarithms (that Kneser's solution some how knows):

\(
\log^{\circ k} \text{tet}_K(z) \to L,L^*\,\,\text{as}\,\,k\to\infty\\
\)

So if you were to attempt to do the inverse iteration to construct this, where its centered around the behaviour,

\(
\log_k(z_k) = z_{k-1}\\
\)

And we create splines between them; this inherently shouldn't work, right? Because eventually there will be a value \( s_j \) such that,

\(
\text{tet}_K(s_j) = z_j\\
\)

But,

\(
\text{tet}_K(s_j-1) = \log \text{tet}_K(s_j) \neq z_{j-1}\\
\)

And this would mean, as your iteration is stable about these periodic points; there should be some sort of discrepancy when you create the spline between the periodic points. As this spline will be stable under the orbits of \( \log_k \) (with the appropriate key); but as we can see; Kneser's solution is NOT stable under any key about any periodic points.

At least, I think that makes sense. Sorry; I'm not the best with Kneser's solution, or the existing code. As I lack much education in computer programming; it's difficult for me to intuit the fatou.gp program.

I apologize if I'm speaking nonsense, lol.

Regards, James.