jaydfox Wrote:Why wouldn't you have a singularity at both fixed points? I didn't realize what you were saying here, but now that I'm looking at it again, I'm not quite sure I agree.

If I have a development of (with non-zero convergence radius) at one fixed point then the coefficients of this development are already uniquely determined and this development must be the regular iteration. In this case we know the other fixed point must have a singularity, because if it wouldnt then the development at the second fixed point would be unique and the regular one and we know that both developments do not yield the same analytic function.

It is as if we have a piece of cloth and if we pull it to be smooth at one fixed point it gets corrugated at the other fixed point(s).

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