the inconsistency depending on fixpoint-selection

[bo198214]

Now I'd like to understand, what is the source of the difference of selection e=1 vs d=1. The most obvious difference was in the given case

Hm, I had another look at your transformations (which indeed puzzled me). I came up with this:

Your first transformation:
, ,
gives the same result as the regular iteration at the upper fixed point, because:
where and
so and is the function with upper fixed point moved to 0.

But now it is easily derivable that the regular iteration , where is the regular iteration of . This has to do with that the regular iteration is the only one which posses the limit .

So we have .

However your second transformation
, ,
has as translation the same translation as in your previous transformation!
Only changes to .

So this second transformation moves also the *upper* fixed point to 0. And we have seen that we can dismiss any multiplicative conjugation with respect to regular iteration. So it has of course the same result (fractional iteration of ) as your first transformation.