04/18/2009, 11:24 AM
(This post was last modified: 04/18/2009, 12:24 PM by Kouznetsov.)

bo198214 Wrote:...Shame! 18 years without advances. There should be a paper about it. Let us submit one right now!

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 at the lower and upper real fixed points, , differ! He, Dean Hickerson and Richard Schroeppel found this arround 1991, however there is no paper about it.

...

bo198214 Wrote:...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 .

The numerical computations veiled this fact because the differences are in the order of .

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bo198214 Wrote:...What about the range of holomorphism of each of the 3 functions you mention?

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.

How about their periodicity? Do they have periods?

Below, for base , I upload the plots of two functions:

which is superfunciton of such that and where .

which is superfunciton of such that and , where ; at least for .

[attachment=480]

In the first plot, the lines

const

const

are shown. Thick curves correspond to integer valuse of p and q.

In the second plot, the lines

const

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 is shown in the same notations. The plot of this difference along the real axis is below:

Dashed:

Thin:

Thick: My approximation for

I suspect, each of functions and 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.