Hi James,
a second read of your question makes me think that my previous reply has not been well to the focus of your question.
Perhaps this is a better one.
The existence of n-periodic points, when looking at the forward iteration \(z \to \exp(z)\), gives rise to the assumption, the fractional interpolation could as well be implemented as it can be done by any two points on one trajectory.
But the backward iteration \(z \to \log(z) + k \omega\) where \(k\) gives the branch-index, between two points \(z_j = \log(z_{j+1}) + K[j+1] \omega\) and \(z_{j-1} = \log(z_{j}) + K[j] \omega\) where \(K[j]<>K[j+1]\) cannot smoothly be interpolated: it were needed that also the jump between the two branchindexes \(K[j]\) and \(K[j+1]\) could be smoothed(*). But that would require the re-definition of the complex logarithm, so I think this can never be done. So this has also consequences for the definition of restrictions of the Schroeder function and of its inverse: I think, the idea of the Schroeder-function as idealized/normed infinite iteration has no place in its mathematical derivation/representation for varying branchindexes of the log resp the varying entries of its associated Devaney-string. The best what we could do, in my opinion, with the Schroeder-function were to engineer it to work for other 1-periodic points/fixed points. But I've no idea how at all a variable branchindex could possibly be thought.
I've experimented with this on base of the most simple non-trivial "key" \(K=[0,0,1]\) and used the fatou.gp and my "polynomial method" concurrently (without any visible difference) focusing the problem of the non-smooth step in the branch of the log between \(z_1\) and \(z_3\). See the following picture (draft only so far)
and perhaps more expressive
We see here the ambiguity of continuously continuing the trajectory from \(z_2\) to and beyond \(z_1\) towards \(z_0\) (more precisely to \(z_0'\) where an adaption of the branchindex was not done) respectively from \(z_2\) to and beyond \(z_0\) towards \(z_1\).
--------------------------------------------
Perhaps we could try to make something out of it, if we connect the horizontal strip in the complex plane to make up a torus of perimeter \(2 \pi\) and see, whether this allows meaningful curves, but a short exercise on such a thing seemed to me to lead simply to nothing and/or chaos...
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(*) in older times I've played around with the idea, whether it is possible to introduce a meaning to *fractional* branchindexes of the log, but with no avail. However don't have the link to the tetration-forum post available (searchable keyword something like "fixpointline" or so with some answer of Henryk, I'll insert it here when I've found it)
See: https://math.eretrandre.org/tetrationfor...hp?tid=422 Ufo:Fixpoint-line(?)
a second read of your question makes me think that my previous reply has not been well to the focus of your question.
Perhaps this is a better one.
The existence of n-periodic points, when looking at the forward iteration \(z \to \exp(z)\), gives rise to the assumption, the fractional interpolation could as well be implemented as it can be done by any two points on one trajectory.
But the backward iteration \(z \to \log(z) + k \omega\) where \(k\) gives the branch-index, between two points \(z_j = \log(z_{j+1}) + K[j+1] \omega\) and \(z_{j-1} = \log(z_{j}) + K[j] \omega\) where \(K[j]<>K[j+1]\) cannot smoothly be interpolated: it were needed that also the jump between the two branchindexes \(K[j]\) and \(K[j+1]\) could be smoothed(*). But that would require the re-definition of the complex logarithm, so I think this can never be done. So this has also consequences for the definition of restrictions of the Schroeder function and of its inverse: I think, the idea of the Schroeder-function as idealized/normed infinite iteration has no place in its mathematical derivation/representation for varying branchindexes of the log resp the varying entries of its associated Devaney-string. The best what we could do, in my opinion, with the Schroeder-function were to engineer it to work for other 1-periodic points/fixed points. But I've no idea how at all a variable branchindex could possibly be thought.
I've experimented with this on base of the most simple non-trivial "key" \(K=[0,0,1]\) and used the fatou.gp and my "polynomial method" concurrently (without any visible difference) focusing the problem of the non-smooth step in the branch of the log between \(z_1\) and \(z_3\). See the following picture (draft only so far)
and perhaps more expressive
We see here the ambiguity of continuously continuing the trajectory from \(z_2\) to and beyond \(z_1\) towards \(z_0\) (more precisely to \(z_0'\) where an adaption of the branchindex was not done) respectively from \(z_2\) to and beyond \(z_0\) towards \(z_1\).
--------------------------------------------
Perhaps we could try to make something out of it, if we connect the horizontal strip in the complex plane to make up a torus of perimeter \(2 \pi\) and see, whether this allows meaningful curves, but a short exercise on such a thing seemed to me to lead simply to nothing and/or chaos...
---------------------------------------------
(*) in older times I've played around with the idea, whether it is possible to introduce a meaning to *fractional* branchindexes of the log, but with no avail. However don't have the link to the tetration-forum post available (searchable keyword something like "fixpointline" or so with some answer of Henryk, I'll insert it here when I've found it)
See: https://math.eretrandre.org/tetrationfor...hp?tid=422 Ufo:Fixpoint-line(?)
Gottfried Helms, Kassel
