Iterating at fixed points of b^x

[jaydfox]

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!

To be more precise:
If we have any fixed point \( a \) of \( \exp \), then the power series \( \tau_{-a}\circ \exp\circ \tau_a \) (where \( \tau_a(x):=x+a \)) is a power series with fixed point 0 and with first coefficient \( \exp'(a)=\exp(a)=a \), particularly the series is non-real. By the previous considerations the fixed point is repelling so we have the standard way of hyperbolic iteration (on the main branch) which yields again a non-real power series. If we however afterwards apply the inverse transformation

\( \exp^{\circ t}:=\tau_a\circ (\tau_a^{-1}\circ \exp\circ \tau_a)^{\circ t}\circ \tau_a^{-1} \)

we get back real coefficients and they do not depend on the fixed point \( a \). Though I can not prove it yet, this seems to be quite reliable.

And of course those coefficients are equal to those obtained by the matrix operator method, which in turn equals Andrew's solution after transformation.

Somehow you lost me here. I don't doubt what you say; I'm just not sure what you meant. Perhaps my problem is that I'm misunderstanding how to apply your tau function. And I'm not entirely sure what you mean by "power series". Are you talking about the coefficients of a Taylor series expansion, or are you talking about a series of successive exponentials (i.e., a series of powers). I think I've possibly been sloppy in my own usage of the term "power series", which is leading to my present confusion.