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

[Gottfried]

(...)
I get that you're describing this using the inverse Schroder function,

\(
z_i = \Psi(L^i \xi)\,\,\text{for}\,\,\xi \in \mathbb{C}\\
\)

And I assume you're trying to construct Kneser's solution; which is usually done about a fixed point, but somehow you mean to do it about a cycle? I'm genuinely curious as to how one would do that. Are we essentially just performing Kneser's method on,

\(
\exp^{\circ n}(z)\,\,\text{about}\,\,z = z_i\\
\)

Whereupon, we know this solution will still be Kneser (as far as I understand Kneser this will happen; correct me if I'm wrong, but can't one show this by uniqueness?).
(...)

Hi James, 

I used the Kneser-solution implemented in Sheldon's "fatou.gp" program, starting at one of the 6 periodic points, say \(z_0\) , and calculating fractional iterates from that point with iteration-height \(h=0..1\) towards \(z_1\) then \(h=1..2\)  towards \(z_2\) and so on.  (If I've it correct at the moment, I had crosschecked the results with the simple  "polynomial method", which uses \(h\)-fractional powers simply of the truncated Carlemanmatrix for \(b^x\)  - which gives often good approximations for the Sheldon's Kneser-solution)                 

I did not see "Periodic points" discussed in our forum so far, and for the forward iteration \(z_{k+1}=b^{z_k}\) it is for most bases not obvious to find periodic points at all, because of the divergent character of the iteration, say with \(b=e=\exp(1)\) for which it is known that there are no attracting fixpoints (1-periodic points) and that n-periodic points of any order are repelling.                  
But if you use the backwards-iteration iteratively, applying the \(\log()\), then that points (1-periodic or n-periodic) are attracting. With iterative application of the \(log()\) for your fixpoint-iteration you only can find the primary fixpoint, and using the \(log()\) at branches you find the secondary fixpoints - which is of course known here in the forum from the very beginning.                 
I extended that iteration by using the branches of the logarithm varying (but cyclic) in the sequence of iterations, so with a "key" of \(K=[0,0,1]\) I do fixpoint iteration two times with \(log()\) at the zero-branch, and the third time at the branch with \(1*(2*Pi*I)\). For the example in the initial posting I used the "Key" \(K=[1,2,2,1,0,0]\) for six consecutive iterations over the branched logarithm until periodic points are approximated well enough. Note, that the forward iteration \(b^z\) "eats" the branching and we see actually the 6-periodic cycle without any obvious "key". (Note moreover, that Devaney has introduced the notion of a "string associated with" the iteration of exp(). and from a short discussion with W. Bergweiler (see references in my treatize) I've learned, that that "string" is exactly what I had introduced and named as "key".)                     
The easiest way to proceed, once one has the fractional interpolation between, say, \(z_1\) and \(z_2\) (in my example above \(p_1\) and \(p_2\) - using the letter \(p\) for (p)eriodic points), is to simply exponentiate the Kneser-interpolated line one time, two times,... 5 times to get the other fractionally interpolated segments by the functional equation. However, we shall notice a blowing up of the curves of partial trajectories - much different to the spirals which occur if we use the 1-periodic (fix-) points and apply the functional equation beginning at some segment of the trajectory between neighboured \(z_h\) and \(z_{h+1}\). While this is surely the "natural" way to complete the full n-cycle I was curious, how Sheldon's implementation would perform, if I simply demand the full 6-cycle iteration by increasing the \(h-\) iteration parameter from \(0\) to \(6\) (in steps of, say 1/10): not bad, really (!), but of course it cannot give a smooth interpolation when change of branch-index occurs while evaluating the trajectory when a change-of-branch occurs "on-the-road".                 
This inherent non-smooth change of branch-index over the logarithmizing can surely not give a meaningfull smooth curve for the fractionally iterated partial trajectory (=I think=), and it seems that singularities/jumps must occur, when one reduces the stepwidths of the iteration to small-enough extension - but I don't have really a deeper analysis of this yet...


It seems to me (and I think it is somehow obvious), that the smoothness of the fractional iteration, for instance of the Kneser-method must break down at that exemplars in the n-cycle, where the key \(K\) of branchindexes contains variation, say \([...,0,1,...]\) or \([...,2,5,...]\) or so, because the Kneser-solution for the interval between two points \(z_1\) and \(z_2\) should be dependent on the equality of branch-indexes.   
Btw. with easier keys \(K\) than that of the example in my opening post, say \(K=[0,0,1]\) in-stead, I found the Sheldon's Kneser-solution remarkable nice, while of course the inconsistency could not really disappear.                     

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See further explanation/discussion in my initial treatize attached to this post (well, it deals prominently with the surprising observation of sheer existence of n-periodic points at all... but explains the method how to find such sets of points more detailed).
I'll moverover shall later in the day add links to the relavant MSE and MO-posts that I had done about this subject.





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