Iterating at fixed points of b^x

[bo198214]

Here is a zoom of the primary fixed point showing its trajectory for \( b=\eta+10\dots \eta+0.05 \). You see that it heads to \( e \) for \( b\to\eta \).
   

There is also a mysterious base where the primary fixed point is exactly \( i \). This is the case for \( i=b^i \) which is clearly satisfied for \( b=e^{\frac{\pi}{2}}\approx 4.8104 \).

Now a zoom of the secondary fixed point showing its trajectory for \( b=11\dots 1.05 \).