Real and complex behaviour of the base change function (was: The "cheta" function)

[bo198214]

In theory, most of the principles should be the same. The trivial singularities would be at all branched locations of log^[n-2](1) instead of of log^[n-2](-1), if I'm thinking this through correctly (which I might not be, it's 4 AM here and I just got up to give someone a lift to the airport).

We can use my formula above (which presents the by \( f_3 \) without branching induced singularities)
\( \log_b^{[n-3]}(\log_b^{[3]}(a)+\frac{2\pi i k}{\ln(b)}) \)

and apply it to \( b=\eta \) and \( a=e \).
As \( e \) is a fixed point of \( \log_\eta \) we get \( \log_\eta^{[n]}(e) = e \) and have the singularities
\( \log_\eta^{[n-3]}(e(1+2\pi i k)) \) while the opposite base conversion has the by Jay determined singularities \( \log_e^{[n-3]}(\pi i (1+2 k)) \).

Here we have the interesting case that the singularities should converge to \( e \) with increasing \( n \), i.e. a real instead of a non-real fixed point. I am however not sure about the behaviour for increasing \( k \) and hope Jay makes some illustrating pictures