Iterating at fixed points of b^x

[bo198214]

However I am not sure in the moment for which area A the first case applies. Surely \( f^{\circ n}(z)\to\infty \) for each \( z>a_2 \) where \( a_2 \) is the upper fixed point and \( f^{\circ n}(z)\to a \) for all other real \( z \).

So then I simply made a fractal of it:
The more green the color is the more iterations \( n \) does it take that \( |f^{\circ n}(z)|>T \) (i.e. the slower it converges to \( \infty \)) and if it needs more than 10 iterations (i.e. probably it does not converge to infinity but to the lower fixed point) then the color is black. I chose \( b=\sqrt{2} \), the presented rectangle is \( [4\dots 12] \times [-3\dots 3] \), I made it with FractalExplorer.
\( T=170 \)
   
\( T=500 \)
   
\( T=1000000000 \)
   
We see that the convergence to infinity is really chaotic.