[MSE] Fixed point and fractional iteration of a map - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- 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 $F$