01/08/2015, 03:02 PM

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/...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 \( F \)

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/...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 \( F \)

MSE MphLee

Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)

S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)