01/26/2010, 06:21 AM
(01/25/2010, 07:30 PM)bo198214 Wrote: In some other post on this forum I mentioned that the regular Abel function/regular iterate of \( f(x)=b^x \) at the lower fixed point for \( b<e^{1/e} \) is probably not holomorphically continuable in \( b \) to \( b=e^{1/e} \). However if one uses the perturbed Fatou coordinates for \( b<e^{1/e} \) and not the regular ones at the lower fixed point, then it (and the resulting fractional iterates) depends holomorphically on \( b \)!
I have to revalidate this statement, but it appears very promising.
Do you have proof that this is the case? As this could mean the regular iteration is the "wrong" way to do tetration, not the "right" one. It would be interesting to compare the graph of tetration at some base, say \( \sqrt{2} \), obtained through the regular iteration, to that obtained through this method, esp. on the complex plane, and also to determine the magnitude of the disagreement between the two at the real axis.
