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 

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

This 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 must 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)*?


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*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][1]

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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, or Newton-iteration starting at the given coordinates. 

[![image][1]][1]


  [1]: https://i.sstatic.net/twUOs.png


  [1]: https://math.eretrandre.org/tetrationforum/showthread.php?tid=1277