03/08/2013, 02:30 PM
(This post was last modified: 03/08/2013, 08:41 PM by sheldonison.)
(03/08/2013, 06:18 AM)mike3 Wrote: Hmm. We can track the two "principal" fixed points in the plane as they go from, say, base 2, to this base via a non-real contour from above (since I think \( \left(\frac{1}{e}\right)^e \) lies on the branch cut from \( (-oo, e^{1/e}] \), and we "conventionally" define such functions to be "continuous from above" (if the cut goes to the right, then it is "continuous from below").). The upper fixed point should be \( L_{+} \approx 0.36787944117144232159552377016146086745 \) and the lower \( L_{-} \approx -0.19574575248807635792808172595109672367 + 1.6911999209105686636060727529631245323*I \)....Hey Mike, nice graph, thanks! It definitely looks like a superexponentially growing sexp function! I had figured out the repelling fixed points sometime yesterday afternoon, and had generated \( b^{(L_{-}+z)} \) from the repelling fixed point you gave. Then I figured out the lower superfunction would have a period=~-2.04784+2.1555i, which visually matches your plot -- that's as far as I got. In between going to infinity, your superfunction visually seems to approach 1/e in a similar way as it would in the upper half of the complex plane for the conjectured merged sexp(z), so that seems where the sexp(z) would stitch together with the superfunction at the real axis. From above with Im(z)>0, the conjectured sexp(z) would approach a two periodic approximation winding around 1/e, \( (\frac{1}{e}+k\exp(\pi i z)) \).
The interesting part is that \( L_{-} \) is not a fixed point on the principal branch of the base-\( \left(\frac{1}{e}\right)^e \) logarithm. Instead it is fixed on the branch with \( k = -1 \), i.e. \( \frac{\log(z) - 2\pi i}{-e} \), where \( \log \) is the principal branch of the natural log.
The superfunction generated from the "lower" fixed point looks like this:
(scale is from -10 to +10 on both axes).
So I'd guess that it should look like that in the lower half-plane.
Not sure what to do about the upper half-plane, though.
I'm not ready to compute such an sexp(z) function; the merged sexp(z) algorithm for complex bases doesn't work directly for bases on the Shell Thron boundary for rationally indifferent cases, because such bases lack a Schroeder function from the upper fixed point. The algorithm could work for irrationally indifferent bases on the Shell Thron boundary, even those arbitrarily close to his base. Anyway, one can conjecture this sexp(z) exists. Now, one can imagine following sexp(z) on the Shell Thron boundary counterclockwise from eta to this base, which has a Pseudo period=2. What happens if we continue all the way around back to eta? Does the Pseudo period approach 1 in the upper half of the complex plane, winding around e, approaching a 1-cyclic approximation of \( (e+k\exp(2\pi i z)) \)? In the lower half of the complex plane would this it approach a different non-primary fixed point of eta? I also suspect/hope that there is some sort of a generic form for expressing superfunctions for rationally indifferent fixed points superfunctions. I'll post more details later. At the very last, I'll post a more accurate approximation than \( (\frac{1}{e}+\exp(\pi i z)) \), for the superfunction for this tetration base in the upper half of the complex plane.
- Sheldon
