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

[bo198214]

In thinking about it, the singularities are trivial to find.

You mean *some* singularities?!

For a=eta, b=e, anywhere that the exp^[n-2](x) is equal to -1, we will have a singularity. The double logarithm of the double exponentiation, in the respective bases, will be 0.

These are the singularities induced by \( f_3 \).

This makes me wonder, then: for any given n, there are singularities near the real line, and as n increases, these singularities get arbitrarily close.

Can you make a picture for those that dont currently sit down with a computer algebra system computing exactly this?

Imho the \( \log^{[n-2]}(-1) \) converges to the upper primary fixed point of \( \exp \). So why should they come arbitrarily close to the real axis?