[Update] Comparision of 5 methods of interpolation to continuous tetration

[Gottfried]

What surprises me is that the shapes are so close to perfect circles.

I wonder how it looks like if the real fixpoints approaches its limits.

In other words what happens if the base gets close to eta ?

Do we get figure 8 shapes instead of circles because of the anticipation of the pair of conjugate fixpoints ?

What happens with functions with 2 or more fixpoints ? do they also have these circles around their fixpoints ? In other words, do they locally ( around the fixpoints ) behave the same as the plot given by gottfried here ?

Why not ellipses ??

Concerning the shape:[please see my updated previous post, I've inserted two new pictures] as the base point \( z_0=1 \) goes to the left then the shape becomes first a distorted oval (picture 2) and later a -left-open ellipsoidic looking shape going to negative infinity to the left (picture 3).

Note also on the real axis, righthand of the fixpoint, the occurence of some "mirror"- or "Reflexion"-points: just for the two additional iterations to the right(=negative height) from the currrently most right point ( \( \text{reflexion}(z_0) = y_0 \approx 1.86300282587 \) the "zero-reflexion" \( \text{reflexion} (z_{-1}) = y_{-1} \approx 2.37147238675802 \) and then "neg infinity-reflexion" \( \text{reflexion}(z_{-2})= y_{-2} \approx 3.29126767409758 \)).

The left part of the curve has then the horizontal lines +-<not-yet-determined-imaginary-value> as limiting asymptote.

Well, I'm really curious too what the shape is with bases nearer to eta...

Gottfried