complex iteration (complex "height")

[Ivars]

As I am not so interested in particular values perhaps you need to stretch your own brain. The formula I used was:
\( f^{\circ t}(x)=\lim_{n\to\infty} f^{\circ -n}(a(1-q^t) + q^t f^{\circ n}(x)) \), \( q=f'(a) \).
where \( a \) is the attracting fixed point and \( f \) is the function to be iterated.

Specialized to our case using the lower fixed point \( a \) we get the base by \( b=a^{1/a} \) and \( f'(a)=(b^x)'(a)=\ln(b)b^a=\ln(b)a=\ln(b^a)=\ln(a) \), so:

\( b[4]z=\exp_b^{\circ z}(1)=\lim_{n\to\infty} \log_b^{\circ n}(a(1-\ln(a)^z) + \ln(a)^z \exp_b^{\circ n}(1)) \)

My brain is already overstretched... I think I need some software that can calculate infinitely many logarithms

Ivars