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

[tommy1729]

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.

Gottfried

Im not sure what you meant by reflexion. I noticed the fixpoint is a red x in your plots and there are also green x's. I guess that relates.
Maybe its about pseudocenters of the pseudocircles or the analogue for ellipses ( pseudofocal points) . It seems these centers move away from the fixpoints. These centers seem to follow the real iterations although probably not linearly.

The idea of circles becoming other shapes gradually , reminds me of pseudoperiodic functions. Afterall its like f(a+2pi i) = f(a) but f(a+1+2pi i) =/= f(a+1). Kinda.

As for the asymptotic to the left I wonder - based on the idea of pseudoperiodic somewhat too - if that +-<not-yet-determined-imaginary-value> is actually 1.3 2 pi i ? From the formula b 2pi i where b is the base for 1<<b<<eta.

Perhaps a bit naive. I need more training.

As for these shapes *very* near the fixpoints I believe they are mainly determined by the first 2 derivatives of the fixpoint.

The shapes *far* away are then just the ln or exp iterations of the others (ln and exp in base b of course) . This seems to make sense.

For instance *near* the fixpoints if the function is well approximated by ( recenter fixpoint at 0 for simplicity ) :

0 + x + (x^2)/15

,then we get a circle approximation since the derivative is 1.

This is because the complex iterations of x around x=0 are also a circle.

Maybe a bit naive but my first impression.

Thanks for the updates.

regards

tommy1729