Hmm, I could verify the fractional heights for h=1/2 , h=1/3, ... h=1/6 by reinserting the values in the according powerseries (using the stirling-transformation). Iterations with h=1/8 reproduce up to h=4/8 = 1/2 correctly, and fail at h=6/8 or h=7/8 .

(This does not neccessarily mean, that the results for the fractional iterates of the previous posting are wrong, but the series may be useless for multiple repeated applications)

With finer fractional heights there seem to be generally trouble which I didn't try to investigate yet. I'm nearly sure this is due to the modulus 2*Pi*I .

In another investigation I took especially care for the effect of the exp-function, which reduces the windings to (mod 2*Pi*I) and got an correct answer where the use of the exp-function led to a wrong result, so I'm beginning to consider whether we should build a library of functions/operators, where the operations keep track of the integer multiples of 2*Pi*I as well. Maybe this will give another improvement for the difficult bases. Don't know, whether I can proceed here...

Gottfried

Gottfried Helms, Kassel