Hi Mike,
I think I've the source of the problem/of my lack of understanding now.
I couldn't accespt, that the well known many-to-one-transformation of the exp, which does not include a one-to-many transformation (multivaluedness) for the integer-values of a (iteration-)height-parameter, should be find a generalization that for such fractional positive delta-heights actually also a one-to-many-relation occurs.
To make this visually I tried to understand the key-issue if I compared two assumptions of real height-trajectories:
a) the pathes of the many-to-one come from some height z(h-1), meet at z(h) and all following z(h+k), but have different continuous trajectories for fractional {dh} at z(h+{dh})
b) after the trajectories meet at z(h) all trajectories follow the same path.
Here is a sketch illustrating a): (rightclick to enlarge)
![[Image: attachment.php?aid=741]](http://math.eretrandre.org/tetrationforum/attachment.php?aid=741)
and here another one illustrating b):
![[Image: attachment.php?aid=742]](http://math.eretrandre.org/tetrationforum/attachment.php?aid=742)
Well, intuitively I meant that some fixed function f°dh(x), which implements the real-height-iteration from some point x=z0 , for instance by a certain powerseries, cannot provide discrete differing outputs or even continuously "smeared" directions if I assume only one real parameter changing, so this would prefer version b) over a). On the other hand, if I got the following corrctly: "two powerseries, which produce the same continuous curve over the same finite range x0..x1 must be the same, thus the two powerseries must be identical - and can not have different values elsewhere" (paraphrased from memory) so it would be a strange adventure to try to find some solution for this model...
To improve my intuition I looked at a simpler example, the iteration of the function f(x) = x^3. For this a simple continuous iteration can be formulated F(x,h) = x^(3^h) . Clearly, applied to negative or even complex values z0 noninteger heights h can produce chaotic values due to the multivaluedness of log. (I'll provide some pictures in a second post). What I discussed was the application of continuous iteration starting at some complex value z0 = 1+ 0.5*I .
Because for any value z we have the three complex sources w0<>w1<>w2 where w0^3 (= w0^^1) = w1^3 (=w1^^1) = w2^3 ( = w2^^1) = z and also w0^^2 = w1^^2 = w2^^2 = z^^1 = z^3 I assumed, the trajectories for fractional heights h beginning at w0,w1 and w2 would combine at z0 and proceed from there jointly. But this is not true - and the functions f(x) and F(x,h) are easy enough to trust the correctness of computation.
The (possibly) best answer is illuminating: if I express the complex w1,w2 in height-differences of the same source w0 then they have complex heights with *different imaginary* parts.
Well, it happens (what I wouldn't have believed) that some complex value (w0) has descendants whose rectangular complex representation matches when iterated to some different complex heights h0, h1 and h2.
But in my example I assume only real height-differences, which means that only the real part of the height parameter changes. And thus the continuous trajectories from different w0,w1,w2 along *real* heights is in general distinct (and may sometimes match).
The essential here seems to be, that delta-h is in fact complex and not only real; it has a "length" and a "direction" , thus two, and not only one parameter - contrary to my intuition, that delta-h is only real.
Well, that all shall need a certain time for me to be taken in more depth and to get the picture for the tetrational function sharper.
I have two more points yet.
Gottfried
I think I've the source of the problem/of my lack of understanding now.
I couldn't accespt, that the well known many-to-one-transformation of the exp, which does not include a one-to-many transformation (multivaluedness) for the integer-values of a (iteration-)height-parameter, should be find a generalization that for such fractional positive delta-heights actually also a one-to-many-relation occurs.
To make this visually I tried to understand the key-issue if I compared two assumptions of real height-trajectories:
a) the pathes of the many-to-one come from some height z(h-1), meet at z(h) and all following z(h+k), but have different continuous trajectories for fractional {dh} at z(h+{dh})
b) after the trajectories meet at z(h) all trajectories follow the same path.
Here is a sketch illustrating a): (rightclick to enlarge)
and here another one illustrating b):
Well, intuitively I meant that some fixed function f°dh(x), which implements the real-height-iteration from some point x=z0 , for instance by a certain powerseries, cannot provide discrete differing outputs or even continuously "smeared" directions if I assume only one real parameter changing, so this would prefer version b) over a). On the other hand, if I got the following corrctly: "two powerseries, which produce the same continuous curve over the same finite range x0..x1 must be the same, thus the two powerseries must be identical - and can not have different values elsewhere" (paraphrased from memory) so it would be a strange adventure to try to find some solution for this model...
To improve my intuition I looked at a simpler example, the iteration of the function f(x) = x^3. For this a simple continuous iteration can be formulated F(x,h) = x^(3^h) . Clearly, applied to negative or even complex values z0 noninteger heights h can produce chaotic values due to the multivaluedness of log. (I'll provide some pictures in a second post). What I discussed was the application of continuous iteration starting at some complex value z0 = 1+ 0.5*I .
Because for any value z we have the three complex sources w0<>w1<>w2 where w0^3 (= w0^^1) = w1^3 (=w1^^1) = w2^3 ( = w2^^1) = z and also w0^^2 = w1^^2 = w2^^2 = z^^1 = z^3 I assumed, the trajectories for fractional heights h beginning at w0,w1 and w2 would combine at z0 and proceed from there jointly. But this is not true - and the functions f(x) and F(x,h) are easy enough to trust the correctness of computation.
The (possibly) best answer is illuminating: if I express the complex w1,w2 in height-differences of the same source w0 then they have complex heights with *different imaginary* parts.
Well, it happens (what I wouldn't have believed) that some complex value (w0) has descendants whose rectangular complex representation matches when iterated to some different complex heights h0, h1 and h2.
But in my example I assume only real height-differences, which means that only the real part of the height parameter changes. And thus the continuous trajectories from different w0,w1,w2 along *real* heights is in general distinct (and may sometimes match).
The essential here seems to be, that delta-h is in fact complex and not only real; it has a "length" and a "direction" , thus two, and not only one parameter - contrary to my intuition, that delta-h is only real.
Well, that all shall need a certain time for me to be taken in more depth and to get the picture for the tetrational function sharper.
I have two more points yet.
- First is, that I played with the curve for z0^^h with z0=1+0.5*I using the sexp, which we looked at earlier in this thread. When developed one more degree of integer height the curve becomes extremely chaotic. Still I suspect, that this chaos is some two-dimensional Runge-oscillation similar to the known Runge-effect in the one-dimensional case if we interpolate functions by approximation using polynomials of increasing orders. Since interpolations for the tetration are often developed using that same paradigm (stepwise extension of finite polynomials to infinite powerseries, regular tetration as I employ that) I have always some "alarm" in some edge of my brain.
So I tried what would happen, if I correct the trajectory between, say, h1=1 and h2=2 a bit, manually, and see what would the resulting curve for h2..h3 look like then. It was impressive, that the double winding actually could be calmed down to a non-winded, (even "shorter") path. Well, I could not yet make a system out of this, and due to the unquestionable points at integer heights there will still be some winding and self-crossing, but I am impressed, that an improvement (?) is possible at all!
Here we see a reduced winding compared to our first curve at this height interval:
and here the detail of the range where I manually adapted the interpolation as found by regular tetration. I used changes at the set of only 8 coordinates, but with a small program I'm confident we could do this at a much finer grid (don't know whether I'll have time next week)
- The second is only a question:
Quote: That makes sense, though, since the tetrational is asymptotic to the regular iteration (and the corresponding one at the conjugate fixed point) in the direction of \( \pm i \infty \).
In which way do you see there an asymptotic, or better(?): by what formula is such an asymptotic existent?
Gottfried
Gottfried Helms, Kassel
