Hi James -
yes, you've got correctly what I meant with the logarithmizing and the "key".
But then you formulate
Hmm, I don't know how I should understand this...
Then initialize, say \(z_1=1+î\) and repeat \(z_{1+3k+3}= \log(\log(\log(z_{1+3k})))+1*C\) (where \(C=2 *\pi *î\)) until convergence.
I find, by this iteration the three 3-periodic-points \(p_1 = z_{1+3n}\),\(p_2 = z_{2+3n}\),\(p_3 = z_{3+3n}\)
This is then the material I look at.
I can then use the Sheldon's Kneser-implementation (or my simpler "polynomial method" for approximation) to compute, say, 99 points on a curve between \(p_1\) and \(p_2\) meaning 101 points with iterationheight differences of 1/100 between each. This is a discrete orbit of the idealized continuous fractional iteration between \(p_1\) and \(p_2\), let's call the vector of data "line12".
Naturally, by the functional equation I should be able to calculate the according fractional iterates between \(p_2\) and \(p_3\) just by using the functional equation by doing \(line23=\exp(line12)\) and then should continue \(line31=\exp(line23)\)
This vectors line12,line23,line32 should then equal the data computed by the tetration-function say \(\tet(p_1,h)\) for the iterationheight \(h=0..3\) in steps of 1/100. I expected, that the Kneser-implementation/"polynomial method" would simply produce that data for the fractional heights.
My thoughts are no further innovative or so. No new method there. I simply observe, that the curves come out funny, if not chaotic, and I try to locate the source of the problem. I can't however deny, that my suspicion grows, that the problem is a principal one - but don't know, and possibly simply some restrictions in our function-definition might be sufficient.
One idea is to problematize the behave of the (Kneser-) \(\tet(p1,h)\) function for *negative* h. So \(\tet(p1,-1)\) should equal \(\tet(p1,+2)\) and \(\tet(p1,-0.01)\) should equal \(\tet(p1,+2.99)\) but this doesn't happen, because for the fractional negative heights, the fractional logarithm "does not know" ;-) which & when it should adapt branch index...
The pictures in my recent post take a simpler "key" than that in my initial post (\(K=[0,0,1]\)) to display things more focused. I can also post the actual data set for the pictures if this would make my considerations more transparent.
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
yes, you've got correctly what I meant with the logarithmizing and the "key".
But then you formulate
(04/30/2021, 05:32 AM)JmsNxn Wrote: 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...
(04/30/2021, 05:32 AM)JmsNxn Wrote: So I'm seeing this as a goal to describe what sequence of logarithms work. Correct me if I'm way off here; I'm just trying to understand.Here is perhaps one source of misunderstanding. No, I just give a "key", say \(K=[0,0,1]\).
Then initialize, say \(z_1=1+î\) and repeat \(z_{1+3k+3}= \log(\log(\log(z_{1+3k})))+1*C\) (where \(C=2 *\pi *î\)) until convergence.
I find, by this iteration the three 3-periodic-points \(p_1 = z_{1+3n}\),\(p_2 = z_{2+3n}\),\(p_3 = z_{3+3n}\)
This is then the material I look at.
I can then use the Sheldon's Kneser-implementation (or my simpler "polynomial method" for approximation) to compute, say, 99 points on a curve between \(p_1\) and \(p_2\) meaning 101 points with iterationheight differences of 1/100 between each. This is a discrete orbit of the idealized continuous fractional iteration between \(p_1\) and \(p_2\), let's call the vector of data "line12".
Naturally, by the functional equation I should be able to calculate the according fractional iterates between \(p_2\) and \(p_3\) just by using the functional equation by doing \(line23=\exp(line12)\) and then should continue \(line31=\exp(line23)\)
This vectors line12,line23,line32 should then equal the data computed by the tetration-function say \(\tet(p_1,h)\) for the iterationheight \(h=0..3\) in steps of 1/100. I expected, that the Kneser-implementation/"polynomial method" would simply produce that data for the fractional heights.
My thoughts are no further innovative or so. No new method there. I simply observe, that the curves come out funny, if not chaotic, and I try to locate the source of the problem. I can't however deny, that my suspicion grows, that the problem is a principal one - but don't know, and possibly simply some restrictions in our function-definition might be sufficient.
One idea is to problematize the behave of the (Kneser-) \(\tet(p1,h)\) function for *negative* h. So \(\tet(p1,-1)\) should equal \(\tet(p1,+2)\) and \(\tet(p1,-0.01)\) should equal \(\tet(p1,+2.99)\) but this doesn't happen, because for the fractional negative heights, the fractional logarithm "does not know" ;-) which & when it should adapt branch index...
The pictures in my recent post take a simpler "key" than that in my initial post (\(K=[0,0,1]\)) to display things more focused. I can also post the actual data set for the pictures if this would make my considerations more transparent.
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
Gottfried Helms, Kassel
