eta as branchpoint of tetrational

[sheldonison]

Lets see what happens, with the fixpoints \( M^\pm(b) \) when moving on the circle \( b=\beta(t)=\eta+ (\eta-\sqrt{2})e^{\pi i t} \) for \( -1<t <1 \) in the next post.

Ok, this happens when moving the base in the lower halfplane \( t\in (-1,0) \). The blue curve is the movment of the upper fixpoint \( M^+(\beta(t)) \) and the red one is the lower fixpoint \( M^-(\beta(t)) \). At t=0 we have the two complex fixpoints with real part around 2.5.

Henryk,

I calculated what happens to the fixed points, in going from circle base \( =2\eta+\sqrt{2} \) to base\( =\sqrt{2} \). Moving along the lower circular path, "underneath" eta, I can seamless morph the upper repelling fixed point 2.478+0.8518i seamlessly to a repelling fixed point of 4.0. But the lower fixed point, which starts out at 2.478-0.8518i, and eventually becomes a fixed point of 2.0, somewhere early along the path, around 7 degees/180, the repelling fixed point of 2.478-0.8518i becomes an attracting fixed point of 2.435 - 0.8239i, before continuing on as an attracting fixed point, moving towards 2.0

Mike has spent more time thinking about and graphing "repelling"/"attracting" fixed point combinations in the complex plane, as well as "repelling"/"repelling" fixed point combinations. I'm relying here entirely on my intuition. When we combine "repelling"/"repelling" fixed points, we get a single horizontal line at real axis (which is no longer real valued), which includes the singularities corresponding to log(0), log(log(0)), log(log(log(0))), just as in the Kneser solution for real sexp(z). The real axis is is where the two functions "mesh" into each other. The upper half of the plane corresponds to the upper fixed point, and the lower half of the plane corresponds to the lower fixed point.

My guess is that when you have an upper repelling fixed point, and a lower attracting fixed point, you still get that horizontal line at the real axis, with its singularities, but you get other horizontal lines all in the lower half of the complex plane, where the singularities at log(0), log(log(0)), log(log(log(0))) show up periodically because of the periodic nature of the attracting fixed point. But this would only be in the lower half of the plane.

My intuition fails me, in knowing what happens to the upper repelling fixed point approaching 4.0, and the lower attracting fixed point approaching 2.0, at the very last step, when we get to b=sqrt(2). It doesn't seem to match the [tex]\text{newsexp}_{\sqrt{2}}(z), that I posted earlier. Perhaps its yet another solution, that is not real valued at the real axis ....

By the way, I believe these complex base sexp functions are exactly what Mike is posting about. In principle, I think I know how to calculate a Kneser mapping pair, with a different theta(z) for each of the functions, where the two theta(z) functions merge to a new combined function. It is likely next on my tetration "todo" list.
- Sheldon