[UFO] Attracting Fixpoints or attracting line?

[Gottfried]

euh ...

what is your intention with this attracting line ?

i was thinking and thinking and thinking about how you would use that for tetration , but i dont see how.

care to explain ?

Actually I do not know exactly.
First -We have this nasty problem, that regular iteration around different fixpoints give different fractional heights - but actually we do not know the difference, only that there occurs some wobbling and the difference vanishes at integer heights.
Now we have some meaningful continuous line between fixpoints - maybe that gives another key for some functional description of that differences.

Second - range of convergence is an important topic. If we use the powerseries for iteration and a point is too far from the fixpoint, we get divergence. So we have the option to use another fixpoint.
Say, base=sqrt(2). Then iterating x0=5 using the schröder-function to implement fractional iteration gives divergence, when the series is developed around the fixpoint 2. But that series developed around the other fixpoint 4 gives nicely converging sums.
Now the fixpoints are separated; if we can use some arbitrary value from the "fixed-line" for the construction of the powerseries, say the complex coordinate nearest to x (x=5 in the example), then -perhaps- we can improve convergence - don't know yet.

The whole idea is just an "UFO" - unidentified floating observation ;-) ; let's see, what we can do with it...

Gottfried