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Why bases 0<a<1 don't get love on the forum?
I like it much to see the iteration-characteristic dependend on the height-parameter. (I still have difficulties to capture the name of "superfunction" instead - is it the Schröder-function? the inverse Schröder-function? The Abel- or the inverse Abel-function?)
So for the naive lurker here the following pictures might be instructive to see some characteristic of the tetration to base=0.01 .

The first picture shows the iteration up to with where is only an epsilon above the lower of the pair of fixpoints which are attracting for the exponentiation to base b. ( ) . This is the sequence of blue points starting from the vertical line going to the right of the picture approximating the attracting fixpoint for the logarithm .

For the intuition I also included the trajectory for another example which is above the upper of the pair of fixpoints () and that trajectory goes to the left meaning the height of the logarithm-iteration goes to the negative, or, the height for the exponentiation-iteration goes to the positive. (the magenta dots). Of course if we extend the iterations in the opposite direction then after two steps we encounter complex values - however, that iterations approach again the logarithm-fixpoint, only through the complex plane and the imaginary part slowly vanishing (which cannot be seen here).

What is nicely visible here, is

a) that the whole range/the whole interior between the two horizontal asymptotes contracts by log-iteration (= height h going to the right side) and expands to the two oscillatory fixpoints 0.013... resp 0.9414... (=height h going to the left side of the picture).

b) And that the range outside the asymptotes (however possibly only the neighbourhood of that interval) contracts oscillatory to the asymptotes by exp-iteration (=height to the left side) and contracts to the fixpoint 0.277... by log-iteration (=height to the right side) ... however by requiring the imaginary/complex numbers (which is however not shown here).


The problem for a smooth interpolation is here, that the consecutive integer iterates oscillate (with the selected value example 1, very good visible at least at the beginning).
Of course one could fit some sinusoidal curve here to interpolate between the integer iterates (the blue dots going to the right).
But it seems more natural to introduce the vertical z-axis here for the imaginary part of numbers, such that we can interpolate by a smooth spiral around the x-axis towards . The introduction of complex values is needed by the iterations to the right of the example-2 anyway).

The following picture shows the interpolation to fractional iteration-heights using the "regular tetration" by the Schröder-mechanism.

Because the fixpoints of the iterated exponentiation are oscillatory I used the attracting fixpoint of the iterated logarithm ( for the conjugation and the definition of the taylor-series , where allows now formally fractional self-composition of its power series. Altogether, this expresses then fractional iterates by fractional iterations of the logarithm.

I used from the inner interval as starting point, computing the fractional iterates of the logarithmizing in steps of 1/20 using that "regular iteration" - machinery.


Finally, to improve the visual imagination once more I tried a 3-D-picture of that "regular iteration" because in the previous picture the axis for the height of the iteration must be the z-axis and is thus not visible in the picture. Here I do a small rotation - however bear with me: it is not a nice waolfram-alpha 3-D-rotation; I just computed the coordinates of the rotated trajectory and not also the reference x/y/z-axis. So it is just a sketch. Anyway, the visualization that I like the most is that of the 3-d-spiral converging like a screw to the fixpoint.


Postscript. Now: we know that the "regular iteration" (via conjugacy and Schröder/Koenigs-function) - irrespectively to its nice conceptual idea and simple computation - has some problems, for instance one would wish to keep the trajectory of complex values with positive imaginary part also to positive imaginary values when iterating with real heights, but we have examples/pictures (see for instace here: ,see discussion & pictures at page 2 & 3 ) where this is not so.

However, the Kneser-like interpolation, which is a development over that "regular interpolation" (as far as I've understood so far) seems to be better in this regard. Computations based on the Kneser-ideas are given above in this thread, however I have not yet values of that Kneser-based interpolation for the given examples; I'd like to see that values inserted in that pictures and how they compete/improve(?) the values from the "regular iteration".

Gottfried Helms, Kassel

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