eta as branchpoint of tetrational

[bo198214]

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.
   

And this happens when moving the base in the upper halfplane \( t\in (0,1) \):
   

The funny thing is, the curves indeed seem to lie on a circle with center 3 and radius 1.