(08/05/2022, 01:20 AM)JmsNxn Wrote: Also, I think this would only qualify as a local iteration if we allowed poles. Where then we are talking about functions on \(\widehat{\mathbb{C}}\). So a local iteration about two points independent of \(t\) will probably be possible, but once extended to its maximal domain will have poles.
Yes, but what I meant, it is a local iteration about *one* fixed point. But, yes, there will never be a domain independent of t that contains *both* fixed points - and this is what I announced the picture for in that post. I call it "pole rush": It shows how periodically the pole rushes through the real line, with increasing t (for c=-0.5).
(08/05/2022, 01:20 AM)JmsNxn Wrote: Also, I don't think you even need to put in as much work as you've done to find a counter example. Taking conjugations of \(\lambda^t z\) by linear fractional transformations, will probably already supply that.Its just that, isn't it? But a very specific one, that shows parabolic plosion on the real axis.