In the concept of fractional iteration of the exponential function ("tetration") the property of $$\exp^{\circ a}(\exp^{\circ b}(z))=\exp^{\circ b}(\exp^{\circ a}(z))=\exp^{\circ a+b}(z) \tag 1$$ has been/is a very basic, even a foundational one, for my understanding.

In some recent consideration of n-periodic points it occured to me, that this cannot hold (independently of the style of interpolation, be it "regular"/"Schroeder-" or "Kneser-" or any other method), where for example with the 3-periodic points $p_1$, $p_2=\exp(p_1)$, $p_3=\exp(p_2)$, $p_1=\exp(p_3)=\exp^{\circ 3}(p_1)$ we have with some point $p_{1.1} = \exp^{\circ 0.1}(p_1)$ the following

$$ p_{1.1}=\exp^{\circ 0.1}(\exp^{\circ 3}(p_1))=\exp^{\circ 0.1}(p_1) \tag {2.1} $$ $$ \text{ but } $$ $$ \exp^{\circ 3}(\exp^{\circ 0.1}(p_1)) \ne p_{1.1} \tag {2.2} $$

Example data: $$ \small { \begin{array} {rll} p_1&=0.90866853431523997218 + 0.67624988547121701164*I \\ p_2&=1.9350078633658531684 + 1.5528005817432416377*I \\
p_3&=0.12459758600926988160 + 6.9229773584312693320*I \\
p_4 &= \exp^{\circ 3}(p_1) = p_1
\end{array} }$$ _{Remark 1: the value of $p_1$ can be made arbitrarily precise by initializing $\small{p_1=1+I}$ and then iterating $\small{p_1=\ln(\ln(\ln(p_1)+2\pi I))}$ to sufficient convergence}

$$ \small{ \begin{array} {rll} p_{1.1} &= \text{rtet}(p_1,0.1) &= 0.99523184831984789219 + 0.70987078655389452780*I \\
p_{4.1 \, a}&=\exp^{\circ 3}(\text{rtet}(p_1,0.1) ) & = 0.048217006014677061506 + 0.22062947724802093650*I \\
p_{4.1\,b}&=\text{rtet}(\exp^{\circ 3}(p_1),0.1)&= 0.99523184831984789219 + 0.70987078655389452780*I \\
p_{4.1\,a} & \neq p_{4.1\,b} \qquad \qquad\text{(!)}
\end{array} }
$$
_{Remark 2: here, rtet(z,height) is an implementation of fractional iteration, which is numerically approximate to the Kneser-solution as given by the implementation of S. Levenstein in the tetrationforum but works simply by diagonalization of a finite truncated Carlemanmatrix of size $\small {16 \times 16}$}

That the two iterations in different orders cannot be equal can easily be seen by the argument,

- that the infinity of
*3*-periodic points (as well as in general*n*-periodic points) is*countable*, - a continuous curve connecting $p_1 \to p_2 \to p_3 \to p_1 \to ...$ if it were itself periodic would represent
*uncountably*many*3*-periodic points,

but which is a contradiction.

Another, perhaps better known, argument comes from the allegory of a "hair" coined by R. L. Devaney in his study of periodic points in the exponential function. It says (paraphrased here)

- that in an epsilon neighbourhood of a periodic point (say $p_1$) the iteration of any point (except of $p_1$ itself) diverges, even chaotically, towards infinity.

Of course this implies as well that there cannot be a trajectory of fractional iterates, whose partial curves $l_{12}=[p_1,p_2]$, $l_{23}=[p_2,p_3]$ and $l_{31}=[p_3,p_1]$ have the 3-periodicity $\exp^{\circ 3}(l_{12}) \ne l_{12}$. (Actually that iterations blow the partial curves out quickly and few such iterations chaotize completely any graphical plot).

This inequality in *(2.1)* breaks the -in my view- foundational equality *(1)* such that I now question at all the meaningfulness of the fractional iteration in such cases.

Do I possibly misrepresent/misinterpret the

*foundational*role of*(1)*?Has tetration been developed so far with well knowing and possibly answering the problem in

*(2.2)*?

*A discussion starting at some initial irritation because of existence of 6-periodic points, the chaotizing of the interpolated trajectories, towards more precise graphical display and towards finding the thoughts presented here can be checked at* tetrationforum

A picture illustrates the partial trajectories and their non-periodicity. The coordinates of the periodic points can be approximated to arbitrary precision by simple fixpoint-iteration: `P1=log(log(log(P1)+2*Pi*I))`

until satisfactory precision (I use, by default, 200 internal decimal digits with Pari/GP), or Newton-iteration starting at the given coordinates.

A short article discussing the initial observation of existence of periodic points is here This does not arrive at the discussion of the problem of non-periodicity of the fractional iterated trajectories along the n-periodic points.

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