Iteration with two analytic fixed points

[Daniel]

So what.
For me there was the conjecture in the space that an iteration can not be holomorphic at two (hyperbolic) fixed points at the same time. Which would boil down to if I have a regular iteration at one point then there is no continuation to the other fixed point. (And it would be totally enough for me to have this statement on the real line only!)

And I simply gave a counter example to that.


I though it was clear from my last sentence that I understood what you would like to have - a vicinity of the fixed point that is independent of t. 

Good, if that helps you with the proof. But I neither saw a proof nor even a fixed conjecture.

Ps: I never claimed equality of regular iteration with what you call local iteration. I did not even mention "local iteration" in my post.

As I have mentioned before, I did a numerical experiment that from a fixed point my method of iteration was able to show where the next fixed point was and what it's Lyapunov multiplier was. If it is important I could try and recreate my experiment.