[AIS] (alternating) Iteration series: Half-iterate using the AIS?

[sheldonison]

....
We take a base, say b=1.3, and use the decremented exponentiation \( f:x \to b^x-1 \), such that we have two real fixpoints....
... If I compute the halfiterate using the regular tetration via the squareroot of the formal powerseries/the Schröder-function mechanism, I get \( x_{0.5(regular)} \sim 0.2273401 704241 \) which is very close, but only to some leading digits. The values of the infinite series beginning at these values differ only by 1e-16 and smaller, so maybe the non-match is an artifact (which I do not believe).

When analyzing iterated exponentiation for a base less than eta, and looking at the behavior between the two fixed points, there is inherent ambiguity, which may or may not be relevant to Gottfried's current example.

Working with b=1.3 for decremented exponentiation, \( f:x \to b^x-1 \), is conjugate to working with tetration for base c=1.3^(1/1.3)=1.22362610172251... \( g:x \to c^x \), where \( f^{[z]} = \frac{g^{[z]}}{1.3}-1 \). So this problem is exactly analogous to looking at tetration base c, \( g:x \to c^x \), which has two fixed points, 1.3 and 12.52457261... \( f:x \to b^x-1 \), also has two fixed points, which are translated by the formula I gave, \( f^{[z]} = \frac{g^{[z]}}{1.3}-1 \), to 0, and 8.634286622...

So then this is tetration for base c, which is less than eta, and it seems similar to Henryk's and Dimitrii's paper on base sqrt(2), which also has two real fixed points. For Gottrfried's current example, we would expect the two real fixed points to lead to slightly different superfunctions, depending on which fixed point is used to generate the Schroeder function. Both superfunctions can be defined to be real valued at the real axis, with values ranging from 0 to 8.634286... but the two have different imaginary periods in the complex plane. From the lower fixed point of zero, with period=4.695878i, we expect a logarithmic singularity, and from the upper fixed point with period=6.775735i, we get super exponential growth. I was curious which of the two superfunctions Gottfried's current algorithm was closer too. It is also possible to have a hybrid of these two superfunctions, which I posted on here, http://math.eretrandre.org/tetrationforu...hp?tid=515.
- Sheldon