Dmitrii Kouznetsov's Tetration Extension

[bo198214]

As for example real regular iteration/tetration is no more possible for \( b>e^{1/e} \) because there is no real fixed point.

I am not sure if I understand you well. At b=2 and b=e, tetration F(z) looks pretty regular (except \( z\le-2 \)), and it is real at z>-2. Complex fixed points are easy to work with.

Oh I use "regular" in the sense of "regular iteration" this is a well studied (mostly by Szekeres and Ecalle) way to compute arbitrary real or complex iterates of a function at a fixed point. There is only one solution for the iterates such that the fixed point still remains analytic or at least asymptotically analytic, this is called regular iteration. You will find the iterational formulas as well the formulas for the coefficients of the powerseries of regular iteration throughout the forum (keywords: hyperbolic and parabolic iteration).

For tetration we have \( b[4]t=\exp_b^{\circ t}(1) \). So if we have a fixed point of \( \exp_b \) then we can just consider \( \exp_b^{\circ t} \) to be the regular iteration, which gives us the regular tetration. In almost all cases the regular iteration at different fixed points give different solutions.

As I now see those regular tetration (at the lower real fixed point, which is btw the only attracting fixed point of \( b^x \)) is cyclic along the imaginary axis: For regular iteration we have the iterational formula:

\( \exp_b^{\circ t}(1)=\lim_{n\to\infty} \log_b^{\circ n}(a(1-c^t) + c^t \exp_b^{\circ n}(1)),\quad c=\exp_b'(a)=\ln(a) \) where \( a \) is the fixed point.

We see that the regular tetration \( b[4](it)=\exp_b^{\circ it}(1) \) is periodic with \( 2\pi/\ln(\ln(a)) \) so it can not have a limit for \( t->\infty \).