Iteration with two analytic fixed points

[bo198214]

There exists an immediate attracting basin \(\mathcal{A}\) about the fixed point \(z=2\). This domain is connected and open.

I am not seeing how this proves that both fixed points can not have the same iteration. Your approach looks a bit like:
We take the attracting basin continue the iteration to this domain and then show that it can not have a repelling fixed point.

But that was clear from the beginning already, the attractive basin can not contain any other fixed points by definition.
What would be needed is to show that there is no way to continue the iteration beyond the attractive basin, i.e. to the repelling fixed point.

I don't want to press this too far Bo, I just hope you can understand that a local iteration is a stricter requirement than Regular iteration; but the two ideas pretty much coincide everywhere.

I thought I emphasized that too much already, and now you ask me whether I understand that? (That it is regular iteration is just by be analytic at the fixed point, that's how regular iteration is defined.)