Bummer!

[Gottfried]

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.

The numerical computations veiled this fact because the differences are in the order of \( 10^{-24} \). I reverified this by setting the computation exactness to 100 decimal digits and using the recurrence formula described here:
\( f^{\circ t}(x)=\lim_{n\to\infty} f^{\circ n}(a(1-r^t) + r^t f^{\circ -n}(x)) \), where \( a \) is the fixed point of \( f \) and \( r=f'(a) \) and \( f(x)=\sqrt{2}^x \).

Henryk -
I'm currently investigating the computation of the different eigenmatrices based on the different fixpoints. Apparently the computations lead to the same tetration-matrices Bs (or Bb) in the examples, where I checked this, and thus to the same coefficients for the exponential series (in column 2 of the constructed Bb-matrix). Now the above seems to say, they are in fact not equal. So I'd like to get more infos about the details of the problem. I could not translate your argument above into my matrix-concept - can you explain a bit more explicite? And: do you know some more arguments for the statement of a difference (unfortunately, Dan Asimov seems to have said, there are no papers available)?

Gottfried