Solving tetration for base 0 < b < e^-e

[mike3]

So I would consider the regular iteration of \( g(x)=f^{\circ 2}(x)=b^{b^x} \) and then just always take the half of the iteration number \( f^{\circ t}(x)=g^{\circ t/2}(x) \).

The interesting thing is that \( g \) not only has the two attracting fixed points \( p_1 \) and \( p_2 \) but also a repelling fixed point in between. Which is attractive for \( g^{-1} \) in the range \( (p_1,p_2) \).

[attachment=551]
Graph for b=0.01 with p1=0.941482102273016 and p2=0.0130925205079953.

So why not do regular iteration at that inbetween fixed point?

Because it's a repelling point, so the resulting superfunction will still be entire, and I want to try and find one with the singularities, etc., as that is more like what is expected from "tetration".