Generalized recursive operators

[GFR]

Concerning the two asymptotes of y = eta[5]x, with eta = e^(1/e):

GFR,
...
The double values returned are also interesting, the positive asymptote value should correspond to some very slow operation- may be inverse pentation, what ever it means-is there a definition?.
How did You show (it can be shown...) that what You have shown ? Analytically?
....

As I said, it can be shown that, for b = eta = e^(1/e) = 1.444667861.., we have two fixpoints in y = b[4]x and that they probably (...) are:
x = {-Pi/2, +Pi/2} = {-1.570796327..., +1.570796327..}, corresponding to two symmetrical horizontal asymptotes of y = eta[5]x, with those values (positive, for x -> +oo, and negative, for x -> -oo). Unfortunately, I was not able (to date) to show it analytically, but only graphically. This explains the word "... probably". See the attachment.

GFR