07/11/2019, 08:16 PM

For there are the following cases:

Actually, I also wanted to show the half iterate in the picture. However it took too long time, so I just explain, how it works.

For the hyperbolic case we have to fixpoints. For each fixpoint there exists an analytic regular solution for fractional iterates that is analytic at that fixpoint. And this solution is not analytic at the other fixpoint. This insight was causing the Bummer thread.

For the parabolic case, which you can imagine as moving the fixpoints together into one fixpoint, there are also two fractional iterates (from left and right) that are (I think its called) asymptotically analytic at the fixpoint.

For the elliptic case, there are two conjugate complex fixpoints and there is a uniqueness criterion for the Abel function, which then can be used to calculate the fractional iterates. The fractional iterates are not analytic at both fixpoints.

Somehow for me its strange that in the hyperbolic and parabolic case there are always two "regular" solutions, while in the elliptic case there is only one "right" solution. Does this indicate that the fractional iterates have are not analytic with respect to the basis b, in ?

Actually, I also wanted to show the half iterate in the picture. However it took too long time, so I just explain, how it works.

For the hyperbolic case we have to fixpoints. For each fixpoint there exists an analytic regular solution for fractional iterates that is analytic at that fixpoint. And this solution is not analytic at the other fixpoint. This insight was causing the Bummer thread.

For the parabolic case, which you can imagine as moving the fixpoints together into one fixpoint, there are also two fractional iterates (from left and right) that are (I think its called) asymptotically analytic at the fixpoint.

For the elliptic case, there are two conjugate complex fixpoints and there is a uniqueness criterion for the Abel function, which then can be used to calculate the fractional iterates. The fractional iterates are not analytic at both fixpoints.

Somehow for me its strange that in the hyperbolic and parabolic case there are always two "regular" solutions, while in the elliptic case there is only one "right" solution. Does this indicate that the fractional iterates have are not analytic with respect to the basis b, in ?