Bummer!

[Kouznetsov]

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the conjecture about the equality of the 3 methods of tetration is shattered.
I received an e-mail of Dan Asimov where he mentions that the continuous iterations of \( b^x \) at the lower and upper real fixed points, \( b=\sqrt{2} \), differ! He, Dean Hickerson and Richard Schroeppel found this arround 1991, however there is no paper about it.
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Shame! 18 years without advances. There should be a paper about it. Let us submit one right now!

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The numerical computations veiled this fact because the differences are in the order of \( 10^{-24} \).
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It is beacuse you stay at the real axis. Get out from the real axis, and you have no need to deal with numbers of order of \( 10^{-24} \).

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So the first lesson is: dont trust naive numerical verifcations. We have to reconsider the equality of our 3 methods and I guess there will show up differences too.

What about the range of holomorphism of each of the 3 functions you mention?
How about their periodicity? Do they have periods?

Below, for base \( b=sqrt{2} \), I upload the plots of two functions:

\( F_{b,4} \) which is \( (\mathbb{C}, 0 \mapsto 3) \) superfunciton of \( \exp_b \) such that \( F_{b,4}(z^*)=F_{b,4}(z)^* ~\forall z \in \mathbb{C} \) and \( F_{b,4}(z+T_4)=F_{b,4}(z)^*~\forall~ z \in \mathbb{C} \) where \( T_4=2\pi i/ \ln(2\ln(2)) \) .

\( F_{b,2} \) which is \( (D, 0 \mapsto 3) \) superfunciton of \( \exp_b \) such that \( F_{b,2}(z^*)=F_{b,2}(z)^* ~\forall z \in D \) and \( F_{b,2}(z+T_2)=F_{b,2}(z)^*~\forall~ z \in D \), where \( T_2=2\pi i/ \ln(\ln(2)) \) ; at least for \( D=\{ z\in \mathbb{C}:~\Re(z)>2 \} \) .


   
   
[attachment=480]
   
In the first plot, the lines
\( p=\Re(F_{b,4}(x+ i y)= \)const
\( q=\Im(F_{b,4}(x+ i y)= \)const
are shown. Thick curves correspond to integer valuse of p and q.

In the second plot, the lines
\( p=\Re(F_{b,2}(x+ i y)= \)const
\( q=\Im(F_{b,2}(x+ i y)= \)const
are shown. Thick curves correspond to integer valuse of p and q.
The dashed lines show the cuts.

On the third plot, the difference \( F_{b,4}(x)-F_{b,2}(x)) \) is shown in the same notations. The plot of this difference along the real axis is below:
   
Dashed: \( y=F_{b,4}(x) \)
Thin: \( y=F_{b,2}(x) \)
Thick: My approximation for \( y=10^{25}(F_{b,4}(x)-F_{b,2}(x)) \)

I suspect, each of functions \( F_{b,4} \) and \( F_{b,2} \) is unique.

P.S. Henryk, could you please help me to handle the sizes of the figures?
I think, the same size would be better.