[MSE] Fixed point and fractional iteration of a map
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.


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

Possibly Related Threads…
Thread Author Replies Views Last Post
  ChatGPT checks in on fractional iteration. Daniel 1 342 10/09/2023, 11:38 PM
Last Post: leon
  Down with fixed points! Daniel 1 371 04/29/2023, 11:02 PM
Last Post: tommy1729
  Bridging fractional iteration and fractional calculus Daniel 8 1,290 04/02/2023, 02:16 AM
Last Post: JmsNxn
  Fractional Integration Caleb 11 2,177 02/10/2023, 03:49 AM
Last Post: JmsNxn
  Iteration with two analytic fixed points bo198214 62 16,588 11/27/2022, 06:53 AM
Last Post: JmsNxn
  Discussing fractional iterates of \(f(z) = e^z-1\) JmsNxn 2 852 11/22/2022, 03:52 AM
Last Post: JmsNxn
  Fibonacci as iteration of fractional linear function bo198214 48 13,105 09/14/2022, 08:05 AM
Last Post: Gottfried
  The iterational paradise of fractional linear functions bo198214 7 2,019 08/07/2022, 04:41 PM
Last Post: bo198214
  Describing the beta method using fractional linear transformations JmsNxn 5 1,854 08/07/2022, 12:15 PM
Last Post: JmsNxn
Question The Different Fixed Points of Exponentials Catullus 22 5,521 07/24/2022, 12:22 PM
Last Post: bo198214

Users browsing this thread: 1 Guest(s)