Bifurcation of tetration below E^-E

[bo198214]

It is also clear that these formulas were obtained taking into consideration y = b[3]y, implying y = b[4]oo

Gianfranco, please stop repeating wrong results. I showed already here that this implication is wrong. The left side has multiple solutions for \( y \) while there can only be one limit \( y \) on the right side. Only one solution of the left side can be the actual limit of the right side.

Then I rephrased your considerations into a more clear mathematical context: What you want to know - if I got this right - is

\( y_0=\lim_{n\to\infty} b[4](2n) \) and
\( y_1=\lim_{n\to\infty} b[4](2n+1) \).

By exchanging \( n \) with \( n+1 \) the limit must remain the same, so we have
\( y_0=\lim_{n\to\infty} b[4](2(n+1))=\lim_{n\to\infty} b[4](2n+2)=\lim_{n\to\infty} b^{b^{b[4](2n)}} = b^{b^{\lim_{n\to\infty} b[4](2n)}} = b^b^{y_0} \)

and similar as above \( y_1=b^{b^{y_1}} \).
We have also the correspondence \( y_1=b^{y_0} \) and \( y_0=b^{y_1} \), as you can easily derive in the above manner.


From there you can apply your formula
\( b=e^{\text{W}(y\ln(y))/y} \)
and find that \( y_0 \) corresponds to the solution when taking the branch 0 of \( W \) and \( y_1 \) corresponds to the solution when taking the branch -1 of W. (or vice versa?)


For the actual computation of \( y \) in dependence of \( b \) it only crosses my mind that:

\( y=b^{b^y} \) is equivalent to \( y^{(1/b)^y}=b \), so if we had a symbol for the inversion of \( f_c(x)=x^{c^x} \) then we had the solution \( y={f_{1/b}}^{-1}(b) \).
\( f_{1/b} \) is a strictly increasing function for \( b\ge e^{-e} \), here the limit case \( b=e^{-e} \):
   
but has 3 (?) solutions \( f_{1/b}(y)=b \) for \( 0<b<e^{-e} \):
   

I would guess the biggest solution is \( y_0 \) and the smallest solution is \( y_1 \) (but what is then the middle solution?).