06/29/2022, 11:04 AM
(This post was last modified: 06/29/2022, 12:24 PM by Gottfried.
Edit Reason: picture inserted
)
Perhaps an additionally illustrating discussion in mathoverflow from 2011:
"Do complex iterations of functions have any meaning?"
1) Question by Anixx (see some graphic of similar style as given here)
2) Answer by me (Gottfried Helms) (as well one picture)
3) A long & detailed (recent! 2022) answer by Tom Copeland . (Also referring some unclear points in my answer of 2011)
=================================================
Here is an excerpt from my answer:
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
quote
This is a picture, where I studied the application of imaginary heights, using the base for exponentiation \(b=\sqrt2\). It has the attracting real fixpoint \(t=2\).
As an example, look at the left side, with \(z_0=1 + 0\cdot î\). Using iteration with real heights (here in steps of \(1/10\) ) we move rightwards to \(z_1=b^{z_0}=b = 1.414...\) and by more iterations more towards the fixpoint \(t =2+ 0 î\). This is indicated by the orange arrows.
Note that because \(t\) is a *fixpoint*, we cannot arrive at points on the real axis more to the right hand!
But using imaginary heights, iterations move from \(z_0 \) to \(z_h\) on the indicated circular curve (computed data are in steps of \(0.1 { \pi \over \ln \beta} î\) see legend), which is indicated by the blue arrow.
This iteration does not go towards the fixpoint, but repeats to cycle around it. On that cycling the trajectory crosses the real axis beyond the fixpoint.
(Legend: the circular curves which connect the computed iteration-values of imaginary heights are Excel-cubic-splines and thus only very rough approximations of the true continuous iterations)
![[Image: aI3Ry.png]](https://i.stack.imgur.com/aI3Ry.png)
end quote
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Perhaps a more expanded, systematic and visual presentation would be good for us all ...
Gottfried
"Do complex iterations of functions have any meaning?"
1) Question by Anixx (see some graphic of similar style as given here)
2) Answer by me (Gottfried Helms) (as well one picture)
3) A long & detailed (recent! 2022) answer by Tom Copeland . (Also referring some unclear points in my answer of 2011)
=================================================
Here is an excerpt from my answer:
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
quote
This is a picture, where I studied the application of imaginary heights, using the base for exponentiation \(b=\sqrt2\). It has the attracting real fixpoint \(t=2\).
As an example, look at the left side, with \(z_0=1 + 0\cdot î\). Using iteration with real heights (here in steps of \(1/10\) ) we move rightwards to \(z_1=b^{z_0}=b = 1.414...\) and by more iterations more towards the fixpoint \(t =2+ 0 î\). This is indicated by the orange arrows.
Note that because \(t\) is a *fixpoint*, we cannot arrive at points on the real axis more to the right hand!
But using imaginary heights, iterations move from \(z_0 \) to \(z_h\) on the indicated circular curve (computed data are in steps of \(0.1 { \pi \over \ln \beta} î\) see legend), which is indicated by the blue arrow.
This iteration does not go towards the fixpoint, but repeats to cycle around it. On that cycling the trajectory crosses the real axis beyond the fixpoint.
(Legend: the circular curves which connect the computed iteration-values of imaginary heights are Excel-cubic-splines and thus only very rough approximations of the true continuous iterations)
end quote
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Perhaps a more expanded, systematic and visual presentation would be good for us all ...
Gottfried
Gottfried Helms, Kassel
