Iterating at fixed points of b^x

[bo198214]

Then I decided to use Mathematica and I got:
7#3 = 3.759823526783788538...x 10^695974 = 3759........2343, an integer number with 695975 figures, covering a printout of about 87 DIN A4 pages.
Therefore, yesterday, I decided to follow a lower profile and tried with a more reasonable and "famous" base, i.e.: rho = 4.810477381... . Well, always using Mathematica, I got:
rho#3 = 6.8101069808199648...X10^1304, which is much more civilized. It is not an integer number, but a DIN A4 page will be enought to show it, with a reasonable precision.
Then, this morning, after my breakfast, I tried to imagine how large could be rho#4 and I had to drink four cups of coffee, to recover. Now (perseverare ... diabolicum) I cannot avoid thinking of what could be rho#100 or rho#1000 or, even, [n->+oo]lim (rho#n) !

Its really fun to read your post.

At the same time, b^x = x (i.e. b^x - x = 0) can be seen as an implicit functional equation, the solutions of which should represent, after infinite iterations, an infinite tetration (infinite tetrates) of base b, which can be written as : y = b#(+oo) = h, the heights of the "infinite tower" with base b.
...
(2) - Kritik of the Mathematical Reason

It is expected that the critical point will be to say that one thing is the limit of rho^n, for n-> oo, and another thing is the determination of the fixpoint of rho^x = x. The countercritical position could be to say that, in this particular case, the two procedures must reach the same non-contradictory results (divine surprise!).

But thats not a critique of *mathematical* reasoning but a critique of *human* reasoning!

Mathematical reason goes like that:
Let \( f \) be a real (or complex) continuous function and suppose that \( y=\lim_{n\to\infty} f^{\circ n}(x_0) \) exists (and this does not include infinity) then \( f(y)=y \).

Proof:
Define the sequence \( x_n:=f^{\circ n}(x_0) \). Then by assumption \( \lim_{n\to\infty} x_n = y \) but we also have by assumption that \( \lim_{n\to\infty} f(x_n)= y \).
And as \( f \) is continuous \( y=\lim_{n\to\infty} f(x_n)=f\left(\lim_{n\to\infty} x_n\right)=f(y) \). q.e.d.

Also the opposite implication, that from \( f(y)=y \) would follow \( \lim_{n\to\infty} f^{\circ n}(x_0)=y \), is not true! Thats just the human temptation to always revert implications.

The apparent overlapping of the two strategies is that also +oo seems to be a fixpoint of y = b^y , in fact b^(+oo) = +oo. But it is not quite so, since b^(+oo) is of an infinite order much larger than +oo. In other words, for any b > eta [eta = e^(1/e)], b^x >> x, for x -> +oo, the real plots of y = b^x and y = x will never cross eachother for x -> +oo (I mean, before x = +oo). So, what! They will cross ... after ?? But, after what?! After ... (I don't dare to say) ... infinite?

Yeah the good old infinity. Unfortunately you can not just simply add infinity to the real or complex numbers (it leads to contradictions).
They only thing I can add are two references (from my other forum) to similar ideas which however are rather in an undeveloped state:
Idea stub 1 which evolved from this thread.
(There is especially for Ivars a thread discussing hyperreals on this forum.)
And idea stub 2.

Please read the attached pdf notes and tell me, if you wish, your reactions. ... Or, please destroy them ... before reading, if you prefer so.

Dunno whether it was a destruction, but I indeed didnt read your attachment yet, but will do it now