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 \( \exp_{b,2}^{[h]}(x) \) and \( \exp_{b,4}^{[h]}(x) \) (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.

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.

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 \( \exp_{b,2}^{[h]}(x) \) and \( \exp_{b,4}^{[h]}(x) \) (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.

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.

Gottfried Helms, Kassel