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[MSE] Fixed point and fractional iteration of a map - Printable Version

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[MSE] Fixed point and fractional iteration of a map - MphLee - 01/08/2015

I've asked this question few days ago on MSE, is about the behaviour of the fractional iterates when there are fixed points. I know that in this forum there was alot of works on the fixed point and stuff, by I have to admit that I can't understand alot.
The better strategy here would be to study some literature and start from 0, anyways at the moment I don't really have alot of time so I started with a very specific question.

http://math.stackexchange.com/questions/1092871/fixed-point-and-fractional-iteration-if-fk-k-then-f1-over-nk-is-anot


The question is the following:


1 - If is a fixed point of the map and ..

2 - exist a map such that (aka behaves as a -iterate of )

prove that is also a fixed point of


On MSE I give a proof but I'm not sure if it is formal, if somone want to try there is a 100 reps bounty there.

PS: If my proof is correct then I guess that even , , .... exc.. are all fixed points of