Hi Jamens -
no nonsense; no. Thanks instead for putting your energy in this!
The shown spline in the first picture was only to give an idea what my problem was, which I could not exactly pinpoint then. It has nothing do to with any thinkable method for interpolation, or, in other words, for a tetration solution - just a sketch to show that I think, the curve of the trajectory of the continuous tetration periodically through the set of the three 3-periodic points would have some problem to be periodic itself.
Now I see, that my problem is much more simple to denote:
let's define the three points \( p_1 \), \( p_2 \), \( p_3 \) which are 3-periodic under \( \exp() \).
Then let's define the point \( p_{1.1}=tet(p_1, 0.1) \) by (any method of) tetration with heights \( h \) over the reals.
Then, one one hand, we'll have \( p_1=tet(p_1,3)=tet(p_1,6)=tet(p_1,9)=... \) which is periodic
but on the other hand we'll have \( p_{1.1}=tet(p_1,0.1)\ne tet(p_1,3.1)\ne tet(p_1,6.1)\ne tet(p_1,9.1)... \) which thus cannot be periodic.
This can also be shown, if we apply the functional equation: \( p_{1.1} \ne \exp^{\circ 3}(p_{1.1}) \) and which is already known elsewhere. In a paraphrasing formulation: "if \(p_1\) is a 3-periodic point, then there are no other 3-periodic points in an epsilon-neighbourhood of \(p_1\)" (which would be required by the assumption that the trajectory of the fractional tetration were as well periodic). It seems, I've just rediscovered the concept of the Devaney's "hairs" where he says, that any point in any small epsilon-neighbourhood of an n-periodic point diverges to infinity when iterated under \( \exp() \). This again means, there are no fractional iterations in the near of \(p_1\) (or \(p_2\) or \(p_3\)) which itself can be periodic.
In a single statement:
The continuous curve, produced by fractional tetration, with increasing iteration-height \(h\) through n-periodic points cannot itself be n-periodic but diverges (chaotically) to infinity for all points except for the n-periodic points themselves (likely paraphrasing Devaney's "hairs" with the here made own observations).
This statement concerns all methods of tetration, be it Kneser, "regular" or whatever.
update: This means also, that the order of iterated exponentiation is no more irrelevant. We have, with a 3-periodic point \( p_1 \), that we get the destruction of the equality
\( \exp^{\circ 3}(\exp^{\circ 0.1}(p_1)) \ne \exp^{\circ 0.1}(\exp^{\circ 3}(p_1)) \) where the rhs is by 3-periodicity \( =\exp^{\circ 0.1}(p_1) \)
, which is what I'd never expected with the construction of the tetration. endupdate
For me, this is catastrophic for the concept of tetration, and maybe this explains also my difficulties to understand the problem at all, and to get a sufficient grip on it, when I got some instinct regarding possible problems in tetration after getting aware of n-periodic points at all in the middle of last year.
What do you think after this more precise focusing of the problem?
If this is not only a trivia (negating/correcting my instinct), I think I should better make a separate post about this...
Kind regards -
Gottfried
no nonsense; no. Thanks instead for putting your energy in this!
The shown spline in the first picture was only to give an idea what my problem was, which I could not exactly pinpoint then. It has nothing do to with any thinkable method for interpolation, or, in other words, for a tetration solution - just a sketch to show that I think, the curve of the trajectory of the continuous tetration periodically through the set of the three 3-periodic points would have some problem to be periodic itself.
Now I see, that my problem is much more simple to denote:
let's define the three points \( p_1 \), \( p_2 \), \( p_3 \) which are 3-periodic under \( \exp() \).
Then let's define the point \( p_{1.1}=tet(p_1, 0.1) \) by (any method of) tetration with heights \( h \) over the reals.
Then, one one hand, we'll have \( p_1=tet(p_1,3)=tet(p_1,6)=tet(p_1,9)=... \) which is periodic
but on the other hand we'll have \( p_{1.1}=tet(p_1,0.1)\ne tet(p_1,3.1)\ne tet(p_1,6.1)\ne tet(p_1,9.1)... \) which thus cannot be periodic.
This can also be shown, if we apply the functional equation: \( p_{1.1} \ne \exp^{\circ 3}(p_{1.1}) \) and which is already known elsewhere. In a paraphrasing formulation: "if \(p_1\) is a 3-periodic point, then there are no other 3-periodic points in an epsilon-neighbourhood of \(p_1\)" (which would be required by the assumption that the trajectory of the fractional tetration were as well periodic). It seems, I've just rediscovered the concept of the Devaney's "hairs" where he says, that any point in any small epsilon-neighbourhood of an n-periodic point diverges to infinity when iterated under \( \exp() \). This again means, there are no fractional iterations in the near of \(p_1\) (or \(p_2\) or \(p_3\)) which itself can be periodic.
In a single statement:
The continuous curve, produced by fractional tetration, with increasing iteration-height \(h\) through n-periodic points cannot itself be n-periodic but diverges (chaotically) to infinity for all points except for the n-periodic points themselves (likely paraphrasing Devaney's "hairs" with the here made own observations).
This statement concerns all methods of tetration, be it Kneser, "regular" or whatever.
update: This means also, that the order of iterated exponentiation is no more irrelevant. We have, with a 3-periodic point \( p_1 \), that we get the destruction of the equality
\( \exp^{\circ 3}(\exp^{\circ 0.1}(p_1)) \ne \exp^{\circ 0.1}(\exp^{\circ 3}(p_1)) \) where the rhs is by 3-periodicity \( =\exp^{\circ 0.1}(p_1) \)
, which is what I'd never expected with the construction of the tetration. endupdate
For me, this is catastrophic for the concept of tetration, and maybe this explains also my difficulties to understand the problem at all, and to get a sufficient grip on it, when I got some instinct regarding possible problems in tetration after getting aware of n-periodic points at all in the middle of last year.
What do you think after this more precise focusing of the problem?
If this is not only a trivia (negating/correcting my instinct), I think I should better make a separate post about this...
Kind regards -
Gottfried
Gottfried Helms, Kassel
