Iterating at fixed points of b^x

[bo198214]

I dont know whether I am the first one who realizes that regularly iterating \( \exp \) at a complex fixed point yields real coefficients! Moreover they do not depend on the chosen fixed point!

I don't think that is true, if I understand you.

Yes I withdrawed this assertion (based on an error in my computation) in the preprevious post.

However I numerically verified (hopefully this time without error) that the regular iterations at both *real* fixed points coincide for \( b<e^{1/e} \).

Just consider the term
\( \ln(a)^n \) in \( \;^{n}b = a + \ln(a)^n \; (1-a) + \ldots \).

Öhm, a bit more explanative?

As a side note, consider any two fixed points \( a_j,a_k \) for the same \( b \), then the fixed point commute under exponentiation \( {a_j}^{a_k}= {a_k}^{a_j} \).

Yes, this is clear as \( a_1^{a_2}=b^{a_1a_2} \) which is independent on the order of \( a_1 \) and \( a_2 \).