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+--- Thread: [MSE] Fixed point and fractional iteration of a map (/showthread.php?tid=947)
[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 \( k \) is a fixed point of the map \( F:X\rightarrow X \) and ..
2 - exist a map \( \Psi:X\rightarrow X \) such that \( \Psi^{\circ n}=F \) (aka \( \Psi \) behaves as a \( 1\over n \)-iterate of \( F \) )
prove that \( \Psi(k) \) is also a fixed point of \( F \)
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 \( \Psi(\Psi(k)) \), \( \Psi^{\circ 3}(k) \), \( \Psi^{\circ 4}(k) \) .... exc.. are all fixed points of \( F \)