This is a tiny question which bothered me from time to time, but I took it always as somehow given.
Now by the current discussions this comes up again. Practically this has a relevance, when I thought to ask my brother, who has a 3-D-printer, for a 3 D model of fractional iteration in the complex numbers, where the height \( h \in \mathbb R \) gives the 3rd dimension, and the integer heights are visualized simply like dots at the floors of the stages in a house. The fractional iterates from one point to another follow then some "noodle" from a point in first floor to the according point in second floor.
(Our chosen interpolation-method define the form of the "noodles".)
The noodles from a fixpoint in the first floor to the same point in the second floor - how is it shaped? Straight, vertical? I remembered I've never consciously seen a discussion of that, I gave it a chance, that it might be simply taken as an axiom. But perhaps, with the obvious informations around from Milnor or Devaney: they might have derived this, or at least might have explicitely mentioned this. (Note, that the "noodles" connecting periodic points have/must have an "exploding", chaotically divergent, form - although I've as well not seen this discussed in the material I have/had made available for me.)
So in this sense:
Gottfried
Now by the current discussions this comes up again. Practically this has a relevance, when I thought to ask my brother, who has a 3-D-printer, for a 3 D model of fractional iteration in the complex numbers, where the height \( h \in \mathbb R \) gives the 3rd dimension, and the integer heights are visualized simply like dots at the floors of the stages in a house. The fractional iterates from one point to another follow then some "noodle" from a point in first floor to the according point in second floor.
(Our chosen interpolation-method define the form of the "noodles".)
The noodles from a fixpoint in the first floor to the same point in the second floor - how is it shaped? Straight, vertical? I remembered I've never consciously seen a discussion of that, I gave it a chance, that it might be simply taken as an axiom. But perhaps, with the obvious informations around from Milnor or Devaney: they might have derived this, or at least might have explicitely mentioned this. (Note, that the "noodles" connecting periodic points have/must have an "exploding", chaotically divergent, form - although I've as well not seen this discussed in the material I have/had made available for me.)
So in this sense:
- are the fractional iterates of a fixpoint "fix" as well/ are they all identically the same coordinate?
- is this taken so-to-say axiomatically, or is it a consequence of something?
Gottfried
Gottfried Helms, Kassel