Tetration Forum

When I was reading Dmitrie's & Henryk's (newest(?)) paper on superfunctions I tried to get an own impression about the differences of tetration when regular iteration is applied with different fixpoints. (see picture 4 at page 22)
I took the base b=sqrt(2) as used in the article and developed at fixpoints a2=2 and a4=4 and considered the range 2<x<4 which can be handled by both functions and (this is the tiny curve in pic 4 on that page)
However I changed the x-axis: instead of equal intervals of x I used equal intervals of h. Now the limits 2 and 4 are at the height-infinities, and there is no inherent center for h=0.
Derived from the graph of differences I used the point x=2.93462 as center-point assigning h=0 to it.
Then I defined the two functions tet2(h) and tet4(h), reflecting the two different fixpoints and plot the differences diff(h) = tet2(h)-tet4(h) for the interval -10 <=h <= 10, which is about 2.04<x<3.96
Here is the picture; the magenta line gives the value of tet2(h) which is between 4 and 2 for -inf<h<+inf, see the y-scale at the right border.

[attachment=501]

There are two aspects which make me headscratching.
(1) I could naively easier accept, if one of the functions proceeds faster and the other one slower; maybe with some modification, for instance a turning point at the center or something like that. But we have permanently changing signs - contradicting the assumtion of a somehow smoothely increasing function. But ok, the behave of the difference can be caused by one of the involved, say by the high (repelling)-fixpoint-version tet4.
(2) But this seems also not to hold. If I assume that at least the tet2-function is smoothely increasing, then a first guess may be, that all differences of all orders should have monotonuous behave. But that's also not true: looking at differences of high order (>24) we find sinusoidal behave in the magnitude of <1e-24. Consequence: very likely also the tet2-function, although based on the attracting fixpoint, has a sinusoidal component in that interval 2<x<4.
[attachment=501]

(08/06/2009 08:56 PM)Gottfried Wrote: [ -> ]When I was reading Dmitrie's & Henryk's (newest(?)) paper on superfunctions I tried to get an own impression about the differences of tetration when regular iteration is applied with different fixpoints. (see picture 4 at page 22)
....
There are two aspects which make me headscratching.
(1) I could naively easier accept, if one of the functions proceeds faster and the other one slower; maybe with some modification, for instance a turning point at the center or something like that. But we have permanently changing signs - contradicting the assumtion of a somehow smoothely increasing function. But ok, the behave of the difference can be caused by one of the involved, say by the high (repelling)-fixpoint-version tet4.
(2) But this seems also not to hold. If I assume that at least the tet2-function is smoothely increasing, then a first guess may be, that all differences of all orders should have monotonuous behave. But that's also not true: looking at differences of high order (>24) we find sinusoidal behave in the magnitude of <1e-24. Consequence: very likely also the tet2-function, although based on the attracting fixpoint, has a sinusoidal component in that interval 2<x<4.

For me, the entire "bummer" post (same topic), and this paper is really interesting. My personal opinion is on the left side, the "smoothest" superexp is developed from the fixed point of four, and on the right side the "smoothest" superexp is developed from the fixed point of two. In between, they're both a little bit "less smooth".

(08/08/2009 04:15 AM)sheldonison Wrote: [ -> ]For me, the entire "bummer" post (same topic), and this paper is really interesting. My personal opinion is on the left side, the "smoothest" superexp is developed from the fixed point of four, and on the right side the "smoothest" superexp is developed from the fixed point of two. In between, they're both a little bit "less smooth".

Hi Sheldon -

well, it may look so, and we may go with it. But I'm not convinced. I'm still thinking, there may be some corrective term in the definition of our fixpoint-dependent functions which let them converge to the same value.
What I've done in continuing the above thoughts was to consider, that the difference on exp_b,2°h(x) and exp_b,4°h(x) also means, that the same value of y must agree to different heights wrt fp2 and fp4. So the sinusoidal form of the above curve should indicate, that at integer iterates the functions synchronize and to the half iterate one function grows faster and to the next half iterate the other one. I've thus plotted this height differences dependent on equidistant height arguments.
Surely we expect periodicity wrt iterates (mod 1), but also the curve seems perfectly sinusoidal. So there seems to be a pair of parameters, the amplitude and a phase-shift to synchronize the two functions, so that we have -like for instance with the zeta-function with positive and negative arguments- a functional relation with different fixpoint-parameter. (The red line is just a mirror of the blue line to make asymmetries better visible)

[attachment=504]
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There is even one more aspect. If you see the related pictures in the bummer thread, then whatever initial x0-point in exp_b,2°h(x0) they use (Dmitri and Andrew used x0=3, I think), then you'll observe, that the x-axis of the sinusoidal is also bended and not vertically symmetric in the middle.

See a sample plot which shows the same curves of differences when I use for x0 values differeing from 3.
(curves have different x0, x-axis is iteration height from there)

[attachment=505]

I could improve vertical symmetry by changing to another x0 (bold red line); which means, that there are points, where the symmetry is even perfect. I optimized that point manually - but to my surprise, there was one possibility to get a (only near?) perfect selection: that was the mirror of x0=1 into that range 2 < x < 4, giving about 2.93
"Mirror" means here: in the schröder-decomposition of the function determine the height-parameter for m0=schröder(x-t), where x is 1, t the fixpoint and u its logarithm.
We'll get for x=1
1=schröder°-1(m0) +t = schröder^-1( u^0*schröder(1-t))+t
Then
x0= mirror(1) = schröder°-1 ( -m0) +t
meaning mirror(1) = b^^(pi*I)
I think, that was something 2.43, and three integer iterates from that was also my manually found value of 2.934...

So if this procedere finds a "normative" value for the phase of the differences, we need another one for the amplitude to define the relation between the two fixpoint-dependent tetrations.



[attachment=505]

(08/06/2009 08:56 PM)Gottfried Wrote: [ -> ](1) I could naively easier accept, if one of the functions proceeds faster and the other one slower; maybe with some modification, for instance a turning point at the center or something like that. But we have permanently changing signs - contradicting the assumtion of a somehow smoothely increasing function. But ok, the behave of the difference can be caused by one of the involved, say by the high (repelling)-fixpoint-version tet4.
(2) But this seems also not to hold. If I assume that at least the tet2-function is smoothely increasing, then a first guess may be, that all differences of all orders should have monotonuous behave. But that's also not true: looking at differences of high order (>24) we find sinusoidal behave in the magnitude of <1e-24. Consequence: very likely also the tet2-function, although based on the attracting fixpoint, has a sinusoidal component in that interval 2<x<4.

No, I dont follow your line of thoughts. Somehow you assume that one of both is the perfect iteration while the other is the bad one with sinoidal deviation, dont you?
Imho both have the same right to live (though only the lower fixed point is usable for a tetrational). Each of them is holomorphic at its fixed point while non-holomorphic at the other fixed point.

I guess with matrix power iteration along the points p in the interval [2,4] we can smoothly deform the tet[p=2] into tet[p=4], where tet[p] is always a superfunction/iteration that though will neither be holomorphic at any of both fixed points.
Remember that matrix power iteration coincides with regular iteration when applied to fixed points.

The oscillating behaviour is not contained in the "bad" tetrational but born through taking the difference. *Both* contribute to the oscillation.
The behaviour must be oscillating as we know that f^{-1}(g(z)) - z is 1-periodic.

(08/11/2009 03:42 PM)bo198214 Wrote: [ -> ]No, I dont follow your line of thoughts. Somehow you assume that one of both is the perfect iteration while the other is the bad one with sinoidal deviation, dont you?

Well, that was a hypothesis from where I was thinking: it may well be, that a function developed at an attracting fixpoint behaves different from one with a repelling one. But I'm (hopefully) not fixed to such hypotheses...

Quote:Imho both have the same right to live (though only the lower fixed point is usable for a tetrational). Each of them is holomorphic at its fixed point while non-holomorphic at the other fixed point.

... which is a good argument...

Quote:I guess with matrix power iteration along the points p in the interval [2,4] we can smoothly deform the tet[p=2] into tet[p=4], where tet[p] is always a superfunction/iteration that though will neither be holomorphic at any of both fixed points.
Remember that matrix power iteration coincides with regular iteration when applied to fixed points.

The oscillating behaviour is not contained in the "bad" tetrational but born through taking the difference. *Both* contribute to the oscillation.
The behaviour must be oscillating as we know that f^{-1}(g(z)) - z is 1-periodic.

Yepp, as I said in my other post: I'll just find it interesting to quantify the difference and possibly have a functional expression for it. (The same problem which I had with the alternating iteration-series ("tetra-series" as I called them then))

(08/11/2009 03:52 PM)Gottfried Wrote: [ -> ]Well, that was a hypothesis from where I was thinking: it may well be, that a function developed at an attracting fixpoint behaves different from one with a repelling one. But I'm (hopefully) not fixed to such hypotheses...

Thats true. Regular superfunctions/functional exponentials can be extended to an entire function if developed at a repelling fixed point. Thats generally not true for one developed at an attracting fixed point, there you can extend it only to a right or left halfplane.

Has anyone done a Fourier analysis of the "wobble"? I tried doing such, and each of the first dozen or so harmonics got about 10^-24 times smaller than the previous, which if the trend holds, would create singularities starting around 9 units off the real axis in the imaginary direction. This fits if we note that the periodicty of the logarithm in the imaginary direction is about 18*I.