08/31/2009, 11:26 AM
(08/30/2009, 04:29 PM)andydude Wrote: This is a really good question. For some reason, I haven't really understood the dependence on the fixed point. I think I remember that if two fixed points are complex conjugates of each other, then the resulting tetrations will also be conjugate, but this is just a consequence of assuming analytic tetration (all analytic functions have this property).yes, just everything is mirrored.
Quote: In the real case (1 < b < eta), I don't see any reason a priori why the resulting tetrations would be different, and my intuition would even want to believe that they are the same, for example that 2 and 4 should produce the same tetrations for b=sqrt(2). Maybe this intuition is wrong, but if it is, then could you point me to the threads in which we talk about this?But Andrew the Bummer thread is all about only this!
Also in the upper superexponential features the both regular iterations at 2 and 4. Or rather there are two at each fixed point which you can see here.
There is even the recognition that the iterations of neraly *every* function is different at different fixed points except the linear fractional, which has two fixed points and the regular iteration is the same at both fixed points.