03/10/2010, 03:10 AM
(03/08/2010, 11:59 AM)bo198214 Wrote: Yes, yes, yes. I am still not that familiar with the theory and technique to give definitive answers. So far it seems these pertubed Fatou coordinates only give real analytic functions for conjugated complex fixed point pairs. To give it a distinguished name (and as "perturbured Fatou coordinates" does not really hit the point) I will call it from here bipolar iteration in mnemonic that the Abel function is defined on a sickel between *two* fixed points, in opposite to the regular iteration which is defined in a neighborhood of *one* fixed point.
First, I dont know yet effective methods to compute these bipolar Fatou coordinates, though I think that Dmitriis algorithm should yield the bipolar superfunction.
Second, the bipolar Fatou coordinates may exist between two real fixed points (though even that is not completely clear to me and not backed by the theory) but it may not be real analytic.
So the question is whether there is at all a real tetration on (1,oo) which is particularly analytic in the base at e^(1/e) ...
Hmm. This suggests the possibility there is an alternative solution instead of regular iteration for the bases in \( (1, e^{1/e}] \) (after all, if it's analytic at \( e^{1/e} \) and the regular isn't, the two can't be equal if they're both analytic in that interval.).
